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THE 



MECHANIC'S, MACHLNIST'S, AND ENGINEER'S 
PRACTICAL 

BOOK OF KEFEKENCE: 



ccntaining tables and formulae for use ix superficial and solid 

mensuration; strength and weight of materials; mechanics; 

machinery; hydraulics, hydrodynamics; marine engines. 

chemistry; and miscellaneous recipes. 



adapted to and for the use of 
ALL CLASSES OF PRACTICAL MECHANICS. 

TOGETHER -WITE THS 

ENGINEER'S FIELD BOOK: 



containing formulae for the various methods of running and changing 
lines, locating side tracks and switches, &c., <&c. 

,. TABLES OF RADII AND THEIR LOGARITHMS, 

^NATURAL AND LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 

NATURAL SINES AND TANGENTS TO EYEET DEGREE 
AND MINUTE OF THE QUADRANT, 

AND 

LOGARITHMS OF NATURAL NUMBERS FROM 1 TO 10,000. 



BY CHARLES HASLETT, 

Civil Engineer. 

EDITED BY CH,ARLES W. HACK LET, 

Professor of Malherrjaticf^ in Columbia College, N. Y. 



N E W Y O R K : 

STRINGER & T O W N S E N D , 222 BROADWAY. 

1 8 5 (i . 



TAi5/ 



10 .H;3^ 



Entered according .to tlie Aoi of Congress, in the year 1855, by 

STKINGEK &z TOWN SEND. 

In the €lerk*s Office of the District Court of the United States for the Southern 
Difitrict of New York. 



In Exchange 

Univ» of Virginia. 
OCT 1 IOC? 



R. CRAIGHEAD, 

.Stereotyper and Electrotyper, 
0:3 Vexeij Street, N. V. 



PREFAC E 



No more useful little works have ever been presented to the 
public than the various pocket companions of a character analogous 
to that here offered. These have been a good deal, though not yet 
too much, multiplied of late ; and where the formulas, rules, and 
tables which they contain have been skilfully framed under the 
guidance of scientific men, they have afforded to the Practical En- 
gineer, Architect, and Mechanic, the most welcome aid in the con- 
structions and computations which make part of their daily occu- 
pation, and which, without the ever-at-hand suggestions and direc- 
tions of these unpretending little servants, might consume hours 
and days in the turning over of large volumes, or in painful in- 
vestigations based on general principles of science where the indi- 
vidual happened to be competent to conduct them. 

The wants to be supplied in such a work are discovered by expe- 
rience and observation in the different callings for which they are 
more especially intended. That these wants have not all been met in 
the works of a similar kind which have already appeared will be 
made evident by a simple inspection of the amount and variety of 
new matter contained in the present volume. 

It is not every one, however practically expert he may be in his 
own pursuit, that is capable of arranging and digesting in the best 
manner the knowledge necessary for his own use which he may 
have been years in acquiring, so as to render it available for the use 



vi Preface. 

I ~~ " ' ~^ 

of others. Such a task, to be well performed, requires a com- 
bination of mental qualities not always, perhaps not often, found 
in the same individual. 

A happy concurrence of circumstances has by accident secured 
for the composition of the present work the labors of several skil- 
ful hands, both as compilers from the best foreign sources, and as 
original producers of valuable material never before in print. The 
result of so much well directed industry is the rich collection, not a 
line of which is not invaluable, which, in the aptest form for imme- 
diate use, has been crowded into the space of a single sinall 
volume. 

Steam and its application play so important a part in the eco- 
nomy of life at the present day, that the most useful practical 
rules and formulas for all the ordinary cases occurring, cannot with 
propriety be omitted in a work of this kind. A due attention will 
be found to have been paid to the matter, and some of the newest 
modes of managing in steam supplied with the means of the requisite 
computation. 

The laying out of Railroad curves is one of the most important 

and at the same time laborious and troublesome duties which the 

Civil Engineer has to perform. So much of this occurring oii every ■ 

j line of Railroad, any, however slight, improvement of method 

! which may serve to facilitate or lessen the labor of tiiis process is a 

; real boon to that large and eminently useful and acconiplished bo^y 

of men to whom the supervision of such operations is comnaitted. 

The use of the more common trigonometric functions, to wit, sines, 
cosines, tangents, and cotangents, which ordinary tables furnish, is 
not well adapted to the peculiar problems which are presented- in 
the construction of Railroad curves. The additional columns of 
secants and cosecants in the tables of Dr. Bowditoh sometimes i 
afford a slight additional facility, which would be much increased 
had we also columns of natural secants as well as logarithmic. 



JWr THE " 

MECHAJilCrS, MACHmiSrS, AND ENGINEER'S 
PRACTICAL 

BOOK OF liEFEllENCE: 



ccntaining tables and formula for use in superficial and solid 

mensuration ," strength and weight of materials ; mechanics ; 

machinery; hydraulics, hydrodynamics; marine engines, 

chemistry ; and miscellaneous recipes. 



ADAPTED TO AND FOR THE USE OF 

ALL CLASSES OF PRACTICAL MECHANICS. 

TOGETHER, "WITH THE 

ENGINEER'S FIELD BOOI: 



CONTAINING FORMULA FOR THE VARIOUS METHODS OF RUNNING AND CHANGING 
I LINES, LOCATING SIDE TRACKS AND SWITCHES, &C., &C. 

I TABLES OF RADII AND THEIR LOGARITHMS, 

! 

NATURAL AND LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, i 



i 



NATURAL SINES AND TANGENTS TO EVERY DEGREE 
AND MINUTE OF THE QUADRANT, 

AND 

LOGARITHMS OF NATURAL NUMBERS FROM 1 TO 10,000. 



BY CHARLES HASLETT, 

Civil Engineer. 

EDITED BY CHARLES W. HACKLET, 

Professor of Mathematics in Columbia College, N, Y. 



N E W Y R K : 

STRINGER & T W N S E N D , 222 BROADWAY. 

1 8 ;3 (J . 



GIFT 

MAR 30 ■-** 

d'JAl 



11 



Entered according to the Ac- of Congress, in the vearlSSa, br 

STEINOEE Sc TOWN SEND. 

In the Clerk's Office of the District Court of the United States for the Southern 
District of Now Tork. 



R. CRAIGHEAD, 

Stereotyper and Electrotype, 
53 Vesey Sir-eet, N. V. 



PREFAC E 



]N"o more useful little works have ever been presented to the 
public than the vai^ious pocket companions of a character analogous • ) 
to that here offered. These have been a good deal, though not yet j 
too much, multiplied of late ; and where the formulas, rules, and ' 
tables which they contain have been skilfully framed under the | 
guidance of scientific men, they have afforded to the Practical En- ! 
gineer, Architect, and Mechanic, the most welcome aid in the con- | 
structions and computations which make part of their daily oocu- ' 
pation, and which, without the ever-at-hand suggestions and direc- 
tions of these unpretending little servants, might consume hours 
and days in the turning over of large volumes, or in painful in- 
vestigations based on general principles of science where the indi- 
vidual happened to be competent to conduct them. 

The wants to be supplied in such a work are discovered by expe- 
rience and observation in the different callings for which they arc 
more especially intended. That these wants have not all been met in 
the works of a similar kind which have already appeared will be 
made evident by a simple inspection of the amount and variety of 
new matter contained in the present volume. 

It is not every one, however practically expert he may be in his ; 
own pursuit, that is capable of arranging and digesting in the best 
manner the knowledge necessary for his own use which he may 
have been years in acquiring, so as to render it available for the use 



v\ Preface. 

of others. Such a task, to be well performed, requires a com- 
bination of mental qualities not always, perhaps not often, found 
in the same individual. 
I A happy concurrence of circumstances has 'by accident secured 
j for the composition of the present work the labors of several skil- 
i ful hands, both as compilers from the best foreign sources, and as 
j original producers of valuable material never before in print. The 
I result of so much well directed industry is the rich collection, not a 
; line of which is not invaluable, which, in the aptest form for imme- 
diate use, has been crowded into the space of a single small i 
volume. 

Steam and its application play so important a part in the eco- 
j nomy of life at the present day, that the most useful practical 
I mles and formulas for all the ordinary cases occurring, cannot with 
I propriety be omitted in a work of this kind. A due attention will 
j be found to have been paid to the matter, and some of the newest 
modes of managing in steam supplied with the means of the requisite 
computation. ,,;: 

The laying out of Railroad curves is one of the most important 
and at the same time laborious and troublesome duties which thj%r 
Civil Engineer has to perform. So much of this occurring on ever^'- 
line of Railroad, any, however slight, improvement of method 
which may serve to facilitate or lessen the labor of this process is a 
real boon to that large and eminently useful and accomplished body, 
of men to whom the supervision of such operations is committed,, f; 
The use of the more common trigonometric functions, to wit, sines, 
cosines, tangents, and cotangents, which ordinary tables fiu'nisli, is 
not well adapted to the peculiar problems which are presented in i 
the construction of Railroad curves. The additional columns of j 
secants and cosecants in the tables of Dr. Bowditch sometimes j 
afford a slight additional facility, which would be much increased 
had we also columns of natural secants as well as logarithmic, j 



pREFACJ':u vii 

Still there would be much labor of computation which may be 

saved by the use of tables of external secants and versed sines, which 

have been employed with great success recently by the Engineers 

on the Ohio and Mississippi Railroad, and which, with the formulas 

and rules necessary for their application to the laying down of 

curves, drawn up by Mr. Haslett, one of the Engineers of that Road, 

; are now for the first time given to the public. This portion of the 

■ volume alone, by the great abridgment of labor for which it pro- 

: vides the means, and the simplicity and convenience of the matter 

; which it furnishes, will give it an extensive circulation among Prac- 

i tical Engineers. 

I But besides this, the Architect, the Shipbuilder, the Mason, the 
; Carpenter, the Joiner, the Manufacturer and Artisan in iron and 
every species of material, will find rules and recipes for all kinds 
of estimates, computations, constructions, compositions, mixtures, 
et cetera, which will excite surprise at their number, novelty, and 
value to every one. 

The contents of this volume are of so varied a nature that it was 
not deemed necessary to make any strenuous efforts to arrange 
them systematically. Being solely intended for a book of refer- 
ence, the relative order of the subjects is immaterial ; and the co- 
pious Table of Contents and Index afford all the assistance that can 
be desired by those who wish to consult its pages. 

Till-: EniToii. 
Columbia College, 

Sept. 185li 



CONTENTS. 



Arithmetical Smxp; 

! Algebraic Symbols, . , 

! FractIcal Geometry, . . 

1. From any given point, in a straight line, lb erect a perpendicular, 

2, 8. Wiien a perpendicular is to be made at or near the end of u given 

line, 

4. To bisect any given angle, 

5. To find the centre of a circle, or radius, that shall cut any three 

given points, not in a direct line, ..... 

6. To find the centre of a given circle, . . . . i . 

7. To find the length of any given arc of a circle, ... 

8. Tlirouijh any given point, "to draw a tangent to a circle, 

9. To draw from or to the circumference of a circle lines tcndin; 

towards the centre, when the centre is inaccessible, 
10, 11. To describe an arc, or segment of a circle, of large radii, 
12. To describe an ellipse, having the two diameters given, . 
18, 14. To describe an elliptic arch, the width and rise of span b-jing 

given, ............. 

15. To describe a parabola, the dimensions being given, 

16. To obtain by measurement the lenajth of any direct line, though 

intercepted by some material object, . 

17. A round piece of timber being given, out of which to cut a beam of 

strongest section, . ... 

18. To measure the distance between two object?, both being inacces- 

sible, 

19. 2). To ascertain the distance, geometrically, of any inaccessibe 

object on an equal plane, 

21. To find the proper j)osit;on for an eccentric, in relation to the o ank 

in a steam-engine, the angle of eccentric rod, and travel of the 
valve, being given, 

22. The throw of an eccentric, and the fruvel of the valve in a steimi- 

engine, also the length of one lever for communicating motion ^'> 
the valve, being given, to determine the proper length for ihe 
other, ....... 

23. To inscribe any regular polygon in a civen circle, . . . . 

24. To construct a squjire upon'a given nght line 

25. To form a- square equal in area to a given triangle 

20. To form a square equal in area to a given rectangle, . . . . 
27. To fttid the length for a rectansrlc wlioso area shall bo equal to that 

of a given square, the breadth of the rectangle being also ^iven, . 

23. To bi'cct any given triangle ". . . 

29 To describe a circle of greatest di(imeter in a given trian;:!e, 
3 ). To form a rectangle of g:'e:itcst surface, in aglVen tri'ingle. 

Cecimal Arithmetic, 

Reduction, 




21 



2\ 
23 1 

2 1 ' 
24 



Contents. 







PAGE 




27 




2T 




28 




28 




28 





Mensuration, 

Mensuration of Supeefioies, 

1. To find the area of any parallelogram, . . . 

2. To find the area of a trapezoid, 

3. To find the area of a triangle, 

4. Any two sides of a right-angled triangle being given, to find 

the third side, . ' . '. . " . .~ . 

5. To find the area of any regular polygon, .... 

Table of multii>liers for polygons from three to twelve sides 

1. The breadth of a polygon given, to find the radius of a ciicle 

to contain that polygon, 

2. The radius of a circle given, to find the length of side, . 

3. The length of side given, to find the radius, . 

6. Having the diameter of a circle given, to find the circumference 

or the circumference given, to' find the diameter, 

7. To find the length of any arc of a circle, .... 

8. To find the diameter of a circle, by having the chord and versed 

sine given, . 

9. To find the area of an ellipsis, or oval, 

10. To find the area of a parabola, or its segment, 

Some of the properties of a circle, 

11. To fnid the area of a sector of a circle, 

12 To find the area of a segment of a circle, . . . 

13. To find the area of a circular ring or space included between two 

concentric circles, 

14. To find the area of an ellipsis, 



30 i 

31 ; 

31 I 
31 I 

31 i 

82 j 

32 I 

r,2 
33 
33 
33 
34 
34 

85 
35 



Mensuration of Solids, 86 

1. To fiikd the convex surface and solid content of a cylinder. . . 36 

2. To determine the dimensions of any cylindrical vessel, whereby j 

to contain the greatest cubical contents, bounded by the least ! 

superficial surface, 36 I 

3. To find the surface and solid content of a con*' or pyramid, . 37 • 

4. To find the surface of the frustum of a cone or pyramid, . . 37 ; 

5. To find the s(;lid content of the frustum of a cone, . . . 37 ! 

6. To find the solid content of the frustum of a pyramid, . . . 38 i 

7. To find the solidity of a wedge, 38 ; 

8. To find the convex surface and solid content of a sphere or globe, 88 ■ 

9. To find the convex surface and solid content of the segment of a 

sphere, ..." 39 

10. To find the solidity of a spheroid, 39 

11. To find the solidity of the segment of a spheroid when the base 

is circular or parallel to the revolving axis, . . . . 40 

12. To find the convex surface and solid content of a cylindric ring, 40 

Decimal Approximations, for facilitating Calculations in Men- 
suration, 41 

Instrumental Aritiiaietic, or Utility of the Slide Rule, . . 42 

Numeration, 42 

To multiply numbers i>y the rule, .... ... 42 

To divide numbers upon the rule, 43 

Proportion, or rule of three direct, 43 

Rule of three inverse, 43 

Square and cube roots of numbers, 43 

The mean proportion between two numbers, 43 

Mensuration of surface, 44 

Tables of gauge points for the engineer's rule and the common slide 

rule, . . . . , ^ 45 

Mensuration of solidity and capacity, 46 

Land surveying, 46 

Power of steam-engines, . 46 

Of engine boilers, "....... 47 



Contents. 



The Laws of MoTio^f,. . 

To find the velocity of a falliug "body at; the end of any number of 
seconds, 

To find the spm^ passed ovir by a faFiingf body in any number of 
seconds^ • . 

To find the space passed over by a falling body when the velocity is 



Rules and formula when a body is acted on by any force, 

Rule for finding the accelerating force of a body, . . . , . 

Time of a body falling down aninclined pfane/ 

The time of oscillation of a simple pendulum, . . . , . , 

Centrifugal force, . , , . . 

The centre of gyration, . , 

The centre of percussion, .......... 

On Work 

To find the units of work in raising a given weight a given height, 

Horse-povv'er, 

Mavj-power, 

Work of animals, 

Traction of horses at various rates of travelling, .... 

Accumulated work, ^ . , . . 

"Work done by machines, 

The liorse-power of an cngiuef ........ 

On the strength of animals, 

To Calciiliite the Different Parts of a Crane as respects Me- 
chanical Advantage, 

Equilibrium and Pressure of Beams — the Parallelogram of 
Forces, 

SpEcrr^TC Gravity, 

Table of specific gravities, < . . . 

Weight of given bulks of water and air, ....... 

The Mechanical Powers, and their Atplication, * . . . 
The Lever, ..... ...*... 

The Pulley, . 

The AVheel and Axle, 

The Inclined Plane, .1 . ^ . ^ . . . . . . 

The Wedge^ 

The Screw, . 

'tooTHED Wheels, ........... 



PAGE 
47 

47 

43 
45 

4^ 
49 I 

r.5 i 

53 I 

54 
54 
55 
53 , 

56 

56 
57 

57 I 



53 



CA 

r.2! 
63 \ 



Dn THE Velocity of Wheels, Drums, Pulleys, &c. 

Steam Power and the Steam- En oint=:, 

To estimate the amount of advantage gained by using iteam expan- 
sively In ft steam-engine, . T 

To calculate the effect of a lever and weight upon the safety-valve of a 
steam boiler, &c., 

To find the proper diameter for a safety-valve, .... 

General proportions of locomotive engines, 

Steam-engine boilers, and their proportions. 

Of the f)ressure of steam, in inches of mercury, j.t different tempora- 
tures, . 

Temperature of steam at different pressures in atmospheres, . 

The elastic force of steam, and corresponding temperature of the wnter 
with which it is in contact, 

The force and temperature of steam in atmosphuros, 

Table of the heating power of various substances, exhibiting the atmost 
quantity of water evaporated by given weights, «fco., 

Nominal horse-power of low pressure eniilnes, 

Nominal horse-power of Idgh pressure engines, 



63 

65 
65 
66 I 

66 1 

67 ! 

G8 \ 

GS j 

'^ I 

73 
75 

76 
T7 



bl 



St 



Contents. 



PAGE 

Proportions of condensing engine? 85 

Revolutions per mile of driving wheels, and consumption of steam and 

water for each sized wheel, 86 

Pressure of steam exclusive of that of the atmosphere, . . . . 87 j 
Pressure exerted by the elastic force of steam at various degrees of 

temperature, . * 88 , 

Amalgams, 89 j 

Varnishes, 89" 

Practical Tables : "Weiget of Metals — Wrought Iron ; 

I Square, Round, and Flat, 97 i 

: Table of Gradients, and Resistance per Ton for each, . . 118 
j Ultimate Breaking Weight in Tons of Cast-Iron Pillars, . .119 

I Table of Strengths of Cast-Iron Shafts, 123 

: Strength of the Teeth of Cast-Iron Wheels, 127 

Method of Ascertaining the Weight of Pipes of Various Me- 

j tals, and any Diameter, 128 

I Weight of Cast-iron Balls, . 1 29 | 

i Table of the Weight of Flat and Rolled Iron, 130 | 

■ Weight of Cast-Iron Pipes, 131 

Table of the Weight of Malleable Iron, 132 

; Dimensions and Weight of Coppers, 1 33 

Weight of Cast-Iron Plates, 133 

Bore and Weight of Cocks; Weight of Lead, Lead Pipe, and 

Copper Tubing, . 134 | 

•Strength of Materials, 135 i 

1 Table of tenacities, resistances to compression, &c., of the common ! 

; materials of construction, . . . . . . . . . 135 j 

: Strength and weight of ropes and chains, 136 ! 

i Resistance to lateral pressure, or transverse action, 136 \ 

I Elasticity and strength of vaiiouS species of timber, .... 136 i 
i Table showing the weight a beam of cast-iron one inch in breadth will 

sustain, . . 137 

Eesistance of bodies to flexure by vertical pressure, .... 141 

Dimensions of cylindrical columns of cast-iron to sustain given loads, . 142 

Elasticity of torsion, 143 

Table of the weight of a superficial foot of plate or sheet iron, copper, 

' and brass, 144 

Comparative weights of different bodies, 145 

The mensuration of timber — ^fliat or broad measure, cubic or solid 

measure, 146 

Cast metal cylinders, 148 

Table of the weight of cast-iron pipes, 148 

Strength of journals of shafts, 149 

^ Table of the diameters of shafts, 151 

Strength of wheels, 152 

' Proportions of wheels, 154 

Alloys, or Miscellaneous Metals, 154 

Mensuration of Circles : Table of the Diameters, Circumfe- 
rences and Areas of Circles, 160 

Table of the circumferences and areas of circles, from 1 to 50 feet, 

advancing by an inch, 167 

Ivory, 174 



Contents. xiii 

! PAGE 

Centre of Attraction, Equilibrium, tfec, . . ... 175 

Cohesion, 17G 

Meciiantcal Laws OF Elastic Fluids, IV*? 

Boyle's or Mario tte's Law, . . 177 

Dalton's and Gay-Lussac's Law, . . 177 

Amonton's Law, 177 

Dalton's Experiment, 178 

Illuminating gases, 178 

Table of Squares, Cubes, Square and Cube Roots of Numbers, 180 

Blacking Recipes, 202 

Strain and Stress of Materials, 204 

Tensile strength of cast iron, 207 

Tensile and compressive strength of wrought iron, 2C8 

Compression of cast iron, 209 

Transverse strength of beams, 2C9 

Transverse strength of cast-iron bars, 211 

Deflection of beams, 212 

Transverse flexure of a wrought-iron bar by horizontal pressure, . .213 

Hollow rectangular beams, ' . . 214 

Experiments on the transverse strength of rectangular tubes of wrought 

iron, 215 

Strength of cast-iron pillars, 21C 

Effects of temperature on the strength of cast ivon, „ . . . 217 
Comparative strength of long pillars; registance to torsion, . , .217 

Strength of ropes, 218 

Procksses for Staining "Woods, . o . . . . .218 

Logarithms : Application and Utility of Common Logarithmic 

Tables, 219 

Table to determine the Distances of the Movable Points in a 

Parallel Motion, 223 

Capillary Attraction, 224 

Woods, to Polish, Preserve, (fee 22G 

Steam Engine: To Estimate the Amount of Effective Power, . 227 

To determine the proper velocity for the piston, 227 

Approximate velocities for pistons, 228 

Parallel motion, 228 

I Properties and Miscellaneous Effects of Heat : with Tables 

i of Expansion and Dilatation of Liquids and Solids, . . 228 

Degrees of the three thermometrical scales, 232 , 

Table of the Weight of Substances of Construction, . . . 234' 
Conducting Power of Materials used in the Construction of 

Houses, 235 

Capacity of Bodies for transmitting Heat, 285 i 

Solders, 285 I 

Table of Proportions for Shafting with Half-lap Couplings, . 23G 

Gradations of Temperature, 230 

Properties of Numrers, 238 

Table of useful numbers, 240 

Surface of Boiler Tubes of different Lengths and Diameters, . 241 



C0NT£NTS. 



! 


PAGE 


Papers, 


342 


Bronzing 


. .... 347 


Enamels, 


348 


Marble Staining, 


349 


Compound Coloes in Dyeing, .... 


350 


Polishes, 


361 


Bookbinders' Recipes, 


352 


Grayons, 


358 


Gilding, 


359 


Glass Stains, 


3Gi 


Factitious Stones, 


352 


I^-KS. 


366 


Waxes, 


369 



THE EXGIXEER'S FIELD-BOOK, 371 

Formulae for running Lines, locating Side Tracks, (fee, . .375 
Table of Chords corresponding to every 100 feet on curve 
FROM 200 to 1000 feet, calculated to every 15 Minutes- 
rate OF curvature, from 15 MINUTES TO 8 DEGREES, RADIUS OF 

1° BEING 5730 FEET, 408 

Table of Radii and their Logarithms, 409 

Table of Chords, . . . 414 

"Eables of Natural and Logarithmic Yersed Sines and Exter- 

;; NAL Secants, ... 415 

ISrATURAL SiNES AND TaNGENTS TO EVERY DEGREE AND MINUTE OF 

THE Quadrant, 457 

Loga;rithms of I^umbers FROM 1 to 10,000, 483 

Table of the Lengths of Circular Arcs, Radius being Unity, . 499 
Explanation of the Uses and Applications of t^e Table of 

Long Chords, 501 

Table of Middle Ordinates, . 503 

Table for Locating Turnouts, 505 

Miscellaneous Notes and Examples, . , 510 

MDEX, 513 



MECaANIC*S, MACHimST'S, AND ENCmEEP.^S 
PRACTICAL 

BOOK OF REFERENCE: 

CONTAINING 

TABLES AND FORMULA 

fOR TJSE IN 

SUPiiEFIClAL AND SOLID MENSURATION; STRENGTH AND 

WEIGHT OF MATERIALS; MECHANICS; MACHINERY: 

HYDRAULICS; HYDRODYNAMICS; MARINE ENGINES; 

CHEMISTRY; AND MISCELLANEOUS RECIPES. 

ADAPTEt) TO AND FOR THfi USfi OP 

ALL CLASSES OP PRACTICAL MECHANICS. 

EDITED BY 

CHARLES W. HACKLEY, 

|8roft«jsor of JKatJemattcs iit Columiia CoHtjit, N. g. 



^il 



THE 



PRACTICAL BOOK OF EEFEEENCE. 



ARITHMETICAL SIGNS. 

The following ^'efinitions of arithmetical signs 
ployed in mechanical calculations, will be found of great value to \ 
those who do not yet understand them, and of some interest to i 
those who are already familiar with their meanings. 

zz: This is the sign of equality, and signifies eqitol to. For exam- 
ple: 12 inches = 1 foot (12 inches is equal to 1 foot). 

4- This is the sign of addition, and signifies plu^, or more. For 
-example : 5 -h S rz 8 (5 added to 3 is equal to 8). 

— This is the sign of subtradioji, and signifies minus, or less. For 
example : 10 — 8 = 2 (10 minus 8 leaves or is equal to 2). 

X Tills is the sign of multiplication, and signifies midtiplied by, 
or into. For example : 10 x 3 = 30 (10 multiplied by 3 is eqital 
^30). 

-J- This is the sign of division, and signifies divided by. For exam- , 
pk ; 156 -T- 6 = 26 (156 divided by 6 is equal to 26) ; or, 24 -f- 4 = 6 * 

24 
(24 Mmded by 4 is equal to 6) ; or — - = 6 (24 fourths are equal to 6 

whol€s). 

: :: : This is the sign of proportion, and signifies proportion, j 

For example: 4 : 6 :; 8 : 12 {as 4 is to 6, so is 8 to 12) ; or 3 : 5 :: i 

3 9 i 

9:15 (that is, as 3 is to 5, so is d to 15) ; ~ =: — . j 

10 

\/ Tliis is the sign of the square root. When it is placed before I 
a number (as thus, y 6 = 25), it means that the square root of thati 
number is required. For example : v 26 = 5, because 5 x 5 = 25 ; 
or, V 9 = 3, because 3 x 3 := 9 ; or, »^ 64 = 8, because 8 x 8 = 64. 

'^ This is the sign of the cube root. When it is placed before a 
number, it means that the cube root of that number is required. ; 
For example: ^ 64 = 4 (that is, 4 x 4 = 16, and 4 x 16 = 64);' 
or, ^ 216 = 6 (that is, 6 x 6 = 36, and 6 x 36 = 216). | 



14 Arithmetical Signs. 



^ When this mark is added to a number (thus, 6'), it means that 
that number is to be squared. For example: 5^= 2o (that is, 
5x5 = 25); or 6^ = 36 (that is, 6 x 6 = 36). 

^ When this mark is added to a number, it means that that num- 
ber is to be cubed. For example; 5^ =5x5x5 = 125 (that is, 
5x5 =25, and 5 x 25 = 125 ; or, 'J^ = 343 (that is, 7 x 7 = 49, 
and 7 X 49 = 243). The index or power (as the small figure an- 
nexed is called) shows how many times a number is to be multi- 
plied by itself. 

This is called the har. It signifies that all the numbers or 

quantities under it are to be taken together. For example: 3 + 5 
X 4 = 32 (3 plus 5 are equal to 8, and tJiat, multiplied by 4, is 

equal to 32); or, 7—3 + 8 = 12 (7 less 3 is equal to 4, and that, 
if added to 8, is equal to 1 2) ; or, 5 x 4 + 3 = 35 (that is, 4 and 3 
are 7, which, if multiplied by 5, is equal to 35); or, 5 x 6 + 4 = 50 
(that is, 6 and 4 are 10, and ten times 5 are 50). The parenthesis 
( ) is sometimes used in place of the bar, thus: (6 + 4) x 5 = 50. 

i ,'. The meaning of this sign is therefore. 

' .' This sign signifies because. 

± The meaning of this sign is perpendicular. 

/_ This sign signifies an angle. 

~ This sign denotes difference, and is placed between two quan- 
tities (as X ^ y) when it is not known which of them is the greater. 

> or ~zi The meaning of these signs is greater than. For ex- 
ample : A B > C D (that is, A B is greater thari C D). 

< or c" The meaning of these signs is less than. For example: 
AB < CD (that is, AB is less than CD). 

• This is a decimal point. When placed before a number (thus, 
.1), it means that that number has a unit (1) for its denominator. 
For example : .1 is the same as ^; .1 25 is the same as yoTo" J -01 is 
the same as yto 5 -^^1 ^^ the same as yttoo > -0001 is the same as 
ToiooS 4:2.85 is the same as 42^^o- ; 57.217 is the same as 57yVoV 

° This is a degree mark. It is written and printed as follows : 
25° (that is, 25 degrees). 

' Tliis is a minute sign. 

" These two accents signify seconds. 

'" These three accents signify thirds. They read thus: 57* It' 
43^' 39'" (that is, 57 degrees, 17 minutes, 43 seconds, and 39 thirds). 



Algebraic Symbols. 15 



ALGEBRAIC SYMBOLS. 

The advantage of these, in a work like the present, may be thus 
illustrated : 

Let / denote the length, h the breadth, and c? the depth of an iron 
beam. If it be desired to express the product of the length and 
breadth, divided by the depth, it is done as follows: 

n 

d 

That is to say, multiplication is expressed b}'- simply- w^riting the 
letters which represent numbers one after the other ; division, by 
drawing a line under the dividend, and writing the divisor below. 
The sum of the length and breadth, divided by the depth, would 
be expressed briefly thus : 

l + h 



The square of the length, multiplied by the cube of the breadth, 
thus : 

The square root of the length, divided by the fourth root of the 
breadth, thus: 

The square root of the difference of the length and breadth, 
divided by the depth, thus : 

d 

The square root of the quotient of the sum and difference of the 
length and breadth, thus : 



/ 



/ + b 
l-h 



Any other letters — as (7, h, c, <fec. — may stand for the given 
dimensions. 

Tliese explanations will serve to jjive the sense of the symbols 
which will be met with throughout tlie work. 



16 



Practical Geometry. 






PRACTICAL GEOMETRY. 



1. From 0.711/ given point, in a straight Imi, to 
erect a perpendicular ; or, to make a line at right 
atigl^s xcith a given Ibie. 

On each side of the point A, from which the' 
line is to be made, take equal distances, as A 6,' 
A c\ and from h and c as centres, with any dis- 
tance greater than h X ov c A, describe arcs cut- 
tins: each other at d\ then will the line A c? be 
the perpendicular requii-ed. 




2. Wlien a perpendicular is to be made at or near 
the end of a givefi line. 

With any convenient radius, and with any dis- 
tance from' the given line A h, describe a portion 
of a circle, as 6 Ac, cutting the given point in A; 
draw, through the centre of the circle n, the Hne 
h n c\ and a line from the point A, cutting the 
intersections at c, is the perpendicular required. 



S. To do the same otherioise. 

From the given point A, with any conve-i 
nient radius, describe the arc dc b\ from d cut 
the arc in c, and from c cut the arc in b ; also 
from c and b as centres, describe arcs cutting 
\ each other in / ; then will the line A ^ be the 
perpendicular as required. 



Xo?e.— When the three side? of a triangle 
respectively, two of the sides faTn a ner • 
the preceding problems, the perpendiculars mav ue conun 
necessanlv requiiecl. 



J 5 equal parts, 
T ^ ich of these nr 

itc u-eiG.v me given Unes. if 




■i. To bisect any given angle. 

From the point A as a centre, -Nvith any radius 
less than the extent of the angle, describe an arc, 
as c (/; and from c and d as centres, describe arcs 
cutting each other at b ; then will the line A b bisect 
the angle as required. 



Practical Geometry. 



11 



5. Tofiiid the centre of a circle, or radius, that 
shall cut any three given points^ not in a direct line. 

From the middle point ^ as a centre, with any 
radius, as 6 c, h d, describe a portion of a circle, 
as c s d', and from r and t as centres, with an 
equal radius, cut the portion of the circle in c s 
and d s ; draw lines through where the arcs cut 
each other ; and the intersection of the lines at 
8 is the centre of the circle as required. 

6. To find the centre of a given circle. 

Bisect any chord in the circle, as A B, by a 
perpendicular, CD ; bisect also the diameter ED 
in/; and the intersection of the lines at /is the 
centre of the circle required. 

7. To find the length of any given arc of a 
circle. 

With the radius A C, equal to Jth the length 
of the chord of the arc A B, and from A as a 
centre, cut the arc in c ; also from B as a centre, 
with equal radius, cut the chord in h ; draw the 
line C h ; and twice the length of the C 

line is the length of the arc nearly. 




c 





8. llirough any given point, to draw a tan- 
gent to a circle. 

Let the given point be at A; draw the 
line A C, on which describe the semicircle 
ADC; draw the line A D B, cutting the 
circumference in D, which is the tangent as 
required. 



9. To draw from or to the circumference of a circle lines tending 
towards the centre, ivhen the centre is inaccessible. 

Divide the whole or any given 
portion of the circumference into 
the desired number of equal parts; 
then, with any radius less than 
the distance of two divisions, de- 
scribe arcs cutting each other, as 
A 1, B 1, C 2, D 2, (fee; draw the 
lines C 1, B 2, D 3, <fec., Avhich lead to tlie centre, as required. 

7b draiv the end lines. 

As A r, F r, from C describe the arc r, and with the radius C 1, 



2* 




18 



Practical Geometry. 




from A or F as centres, cut tlie former arcs at r, or r, and the lines 
Ar, F r, will tend to the centre as required. 

10. 2h describe an arc, or sequent of a circle, of large radii. 

Of any suitable material, construct a triangle, as A B C ; make 

A B, B C, each equal 
in length to the chord 
of the arc DE, and 
height, twice that of 
the arc B b. At each 
end of the chord D E 
fix a pin, and at B, in the triangle, fix a tracer (as a pencil), move 
the triangle along the pins as guides ; and the tracer will describe 
the arc required. 

11. Or otherwise. 

Draw the chord A C B ; aleo draw the line H I) I, parallel with 
the chord, and equal to the height of the segment ; bisect the chord 

in C, and erect the perpen- 
1 dicular C D ; j oin A D, DB ; 
draw A H perpendicular to 
A D, and B I perpendicular 
to B D ; erect also the per- 
pendiculars A 7?, B ^ ; divide 
A B and H I into any number of equal parts ; draw the lines 1 1, 
2 2. 3 a, (fee. ; likewise divide the lines A ??, B n, each into half the 
number of equal parts ; draw lines to D fi-om each division in the 
lines A n, B n, and, through where they intersect the former lines, 
describe a curve, which will be the arc or segment required. 

12. To describe an ellipse^ having the tioo diameters given. 

On the intersection of the two diameters as a centre, with a 
i radius equal to the difference of the semi- 

diametei*s, describe the arc a b ; and from 
6 as a centre, with half the chord b c a, de- 
scribe the arc c d; from o, as a centre, with 
the distance o d, cut the diameters in dr^ 
dt; draw the lines r, .^ s, and t, s, s; then 
from r and t describe the arcs s, s, s, s; also 
r from d and d, describe the smaller arcs 

5, s, 5, s, which will complete the ellipse as required. 

13. To describe an elliptic arch, the width and rise of span being 

'mn. 

Bisect with a line at right angles the chord or span AB; erect 





I 



Practical Geometry. 



m 





the perpendicular A q, and draw the 
line ql) equal and parallel to AC; bi- 
sect A C and A ^ in r and n ; make C I 
equal to CD, and draw the line Irq; 
draw also the line nsD; bisect sD with 
a line at right angles, and meeting the 
line CD in"^; draw the line^^, make 
CP equal to C ^, and draw the line 
(;¥ i; then from ^ as a centre, with the 
radius g D, describe the arc sD i; and 
from k and P as centres, with the ra- 
dius A k, describe the arcs A s and B ^, which completes the arch as 
required. Or, 



14. Bisect the chord AB, and fix at right angles any straight 
guide, as /; c ; prepare, of any suitable 
material, a rod or staff, equal to half 
the chord's length, as def; from the 
end of the staff, equal to* the height 
of the arch, fix a pin e, and at the 
extremity a tracer/; move the staff, 
keeping its end to "the guide and the 
fixed pin to the chord, and the tracer 
will describe one half the arc re- 
quired. 

15. To describe a parabola, the dijnensions being given. 

Let A B equal the length, and C D the breadth of the required 
parabola ; divide C A, C B into any number of equal parts ; also 
divide the perpen- « « A 

diculars Aa and B6 
into the same num- 
ber of equal parts ; 
then from a and 6 
draw lines meeting each division on the Ime A C B ; and a curve 
line drawn through each intersection Avill form the parabola re- 
quired. 

16. To obtain by measurement the length of any direct line, though 
intercepted by some material obj<*ct. 

Suppose the distance between A and B is required, but the right 
line is intercepted by the object C. On the point d, witli any con- 
venient radius, describe the arc c c, make the arc twice the radius in 
length, through which draw the line dee; and on e describe another 





20 



Practical Geometry. 




arc equal in length to once 
the radius, as eff\ draw 
the line efr equal to efd\ 
on r describe the arc jj, 
in length twice the radius ; 
continue the line through 
rj, which will be a right 
line, and de, or e r, equal 
the distance between dr, 
by which the distance between A and B is obtained as required. 

A round piece of timber being given, out of which to cut a beam 
^C of strongest section. 

Divide into three equal parts any diameter in 
the circle, as A c?, e C ; from d or e, erect a per- 
pendicular meeting the circumference of the 
circle, as c?B; draw AB and B C, also AD 
equal to B C, and DC equal to AB, and the 
rectangle will be a section of the beam as re- 
quired. 

18. To measure the distance between 
two objects, both being inaccessible. 

From any point C draw any line C c, 
and bisect it in D; take any point E 
in the prolongation of A C, and draw 
the line E e, making D e equal to D E ; 
in like manner take any point F in the 
prolongation of B C, and make D/ 
equal to F D. Produce A D and e c till 
they meet in d, and also BD and/c 
till they meet in b ; then a b equal A B, 
or the distance between the objects as 
required. 

19. To ascertain the distance, geometrically, 
of any inaccessible object on an equal plane. 

Let it be required to find the distance be- 
tween A and B, A being inaccessible; pro- 
duce the line in the direction of A B to any 
point, as D ; draw the line D c? at any angle 
to the line \ B ; bisect the line D d, through 
which draw the line B 7>, making cb equal 
to Be; draw the line db a; also through c, 
in the direction c A, draw the line acA, in- 
tersecting the line dba; then b a equal B A, 
the distance required. 




pEACTicAL Geometry. 



n 



20. Otherwise, 

Prolong A B to any point D, making 
B C 'equal to C D ; draw the line D a at 
any angle with I) A, and the line C b 
similar to B c ; draw also the line D E F, 
which intersects the line B a ; then a b 
equal B A, or the distance required. 





21. To find the proper position for an 
eccentric, in relation to the craiik in a steam 
engine^ the angle of eccentric rod, and tra- 
vel of the valve, beiyig given. 

r>raw the right line A B, as the situation of the crank at com- 
mencement of the stroke; draw also the line Cc/, as the proper 
given angle of eccentric rod with the crank; then from C as centre, 
describe a circle equal to the travel 
of the valve ; draw the line ef at 
right angles to the line C a, draw 
also the lines 1 1, and 2 2, parallel 
to the line ef; and at a distance 
from ef on each side, equal to the 
lap and lead of the valve, draw the 
angular lines CI, C 2, which are the 
angles of eccentric with the crank, 
for forward or backward motion, as may be required. 



22. The throw of an eccentric, and the travel of the valve in a steam- 
engine, also the length of one lever for communicating motion to the 
valve, being given, to determine the proper length for the other. 

On any right line, as A B, describe a circle A D, equal to the 
throw of eccentric and travel 
of valve ; then from C as a 
centre, with a radius equal 
to the length of lever given, 
cut the line A B, as at d, on 
which describe a circle, equal 
to the throw of eccentric or 
travel of valve, as may be 
required ; draw the tangents B a, B a, cutting each other in tlie 
line AB, and </ B is the length of the lever as required. 



Nb/e— The throw of nn eccentric is equal to the sum of twice 
the distance between the centres of forntiatioa and revolution, as 
ah, or to the degree of eccentricity it is made fo describe, as cd. 
And 

The travel of a valve is equal fo the sum of tlie widths of the two 
steam oF)enin(f», and the valve's excess of length more than just 
sufficient to cover the openings. 





PiiACTiCAL Geometry. 




23. To inscribe any \ 
circle. 



egular polygon in a given I 




Divide any diameter, as A B, into so many ■ 
equal parts as the polygon is required to have 
sides; from A and B as centres, with a radius' 
equal to the diameter, describe arcs cutting! 
each other in C ; draw the line C D through ! 
the second point of division on the diameter e, | 
and the line D B is one side of the polygon re 
quired. 



24. To comitruct a square upon a given 
right line. 

From A and B as centres, with the ra- 
dius A B, describe the arcs A <? 5, Bet/, 
and from c, with an equal radius, describe 
the circle or portion of a circle e d, A B, 
h € ; from h d cut the circle at e and c ; 
draw the lines A ^, Be, also the line st^ 
which completes the square as required. 



25. To form a square equal in area to a given tria7igle. 

Let A B C be the given triangle ; let 
fall the perpendicular B d, and make A e 
half the height dB; bisect e C, and de- 
scribe the semicircle e n C; erect the per- 
pendicular A s, or side of the square, then 
A s tx is the square of equal area as re- 
quired. 

26. To form a square equal in area to a given rectangle. 

Let the line A B equal the length and breadth of the given rec- 
tangle ; bisect the line in e, and describe 
the semicircle A D B ; then from A with 
the breadth, or from B with the length, 
of the rectangle, cut the line A B at C, 
and erect the perpendicular C D, meeting 
the curve at D, and C D equal a side of 
the square required. 



2'7. To find the length for a rectangle whose area shall he equal to 
that of a given square^ the hreadih of the rectangle being also given. 

Let A B C D be the given square and D E the given breadth of 





Practical Geometry. 



23 



rectangle ; continue the line B C 
to F, and draw the line I) F ; also 
continue the line D C to g, and 
draw the line A^ parallel to D F ; 
from the intersection of the lines 
at g, draw the line g d parallel to 
D E, and E d parallel to D ^ ; then 
^Ddg is the rectangle as re- 
quired. 



28. To bisect any given triangle. 

Suppose ABC the given triangle ; bisect one 
of its sides, as A B in e, from which describe the 
semicircle A r B ; bisect the same in r, and 
from B, with the distance B r, cut the diameter 
A B in V \ draw the line v y parallel to A C, 
which will bisect the triangle as required. 



29. To describe a circle of greatest 
diameter in a given triangle. 

Bisect the angles A and B, and draw 
the intersecting lines AD, B D, cut- 
ting each other in D ; then from D as 
centre, with the distance or radius D C, 
describe the circle C ef\ as required. 



30. To form a rectangle of greatest surface^ in a 
given triangle. 

Let A B C be the given triangle ; bisect any 
two of its sides, as A B, B C, in e and d\ draw 
the line e d\ also, at right angles with the line e d, 
draw the lines e p, dp, and ep pd is the rectangle 
required. 






RATIO OF THE HARD:??ESS OF METALS* 



1. Iron, 

2. Platina, 

3. Copper. 



4. Silver, 

5. Gold. 



6. Tin, 

7. Lead. 



STRENGTH OF WOOD. 



All woods are from 7 to 20 times stronger Iransvcrsel}- than lon- 
gitudinally. They become stronger botli ways when dry. 



24 Decimal Arithmetic. 



DECIMAL AEITHMETIC. 

l>ecinial Arithmietic is the most simple and explicit 

mode of performing practical calculations, on account of its doing 
away with the necessity of fractional parts in the fractional form, 
thereby reducing long and tedious operations to a few figures 
arranged and worked in all respects according to the usual rules; 
of common arithmetic. | 

Decimals simply signify tenths ; thus, the decimal of a foot is the ; 
tenth part of a foot, tlie decimal of that tenth is the hundredth of a i 
foot, the decimal of that hundredth is the thousandth of a foot, and 
so might the divisioiis be carried on and lessened to infinity : but in 
practice it is seldom necessary to take into account any degree of 
less measure than a one-hundredth part of the integer or whole 
number. And, as the entire system consists in supposing the whole, 
number divided into tenths, hundredths, thousandths, <fec., no pecu- ' 
liar notation is required, otherwise than placing a mark or dot to \ 
distinguish between the whole and any part of the whole, thus : 
34.25 gallons signify 34 gallons, 2 tenths, and 5 hundredths of a 
gallon; 11.04 yards signify 11 yards and 4 hundredths of a yard; 
16.008 shillings signify 16 shillings and 8 thousandth parts of a; 
shilling ; from which it must appear plain that ciphers on the right ■ 
hand of decimals are of no value whatever, but placed on the left I 
hand they diminish the decimal value in a tenfold proportion : for j 
.6 signify 6 tenths; .06 signify 6 hundredths; and .006 signify 6j 
tiiousandths of the integer or whole number. \ 

Reduction. 

Reduction means the converting or changing of vulgar fractions i 
to decimals of equal value ^ also finding the fractional value of any i 
decimal given. | 

Rule 1. Add to the numerator of the fraction SLuy number of I 
ciphers at pleasure, divide the sum by the denominator, and the ! 
quotient is the decimal of equivalent value. i 

Bule 2. Multiply the given decimal by the various fractional '[ 
denominations of the integer, or whole number, cutting off from the 
right hand of each product, for decimals, a number of figures equal 
to the given number of decimals, and thus proceed until the lowest 
degree, or required value, is obtained. 

JEx. 1. Required the decimal equivalent, or decimal of equal 
value, to y^2 ^^ ^ ^<^^^- 

— ^- = -25, the decimal required. 

HJx. 2. Reduce the fraction i of an inch to a decimal of equal 
vakie. r^o£ ^ .^^o, the decimal required. 



Decimal Arithmetic. 25 



Ex. 3. What is the decimal equivalent to J of a gallon ? 

1:111 — •S'ZS, the decimal equivalent. 

Ex. 4. Required the fractional value of the decimal .40625 of an 
inch. 

j -40625 
i Multiply by i 8 



V. 2 1 9 



3-25000 
2 

To'ooo 



IK . 

rOOOOO |- and 3*2 of an inch, the value required. 



Ex. 6. What is the fractional valua of -625 of a cwt. ? 

•625 

Multiply by 4 qrs 4 

2-500 

X 28 lbs. 28 

14-000 nr 2 quarters and 14 lbs., the value 
~ required. 

Ex. 6. Ascertain the fractional value of "875 of an imperial 
gallon. 

-875 
Multiply by 4 quarts 4 

3-500 
X 2 pints 2 

1-000 = 3 quarts and 1 pint, the value re- 
• required. 

Ex. 7. What is the fractional value of -525 of a £. sterling? 

•525 
Multiply by 20 sh. 20 

10-500 
X 12 pence 12 

6-000 = 10 shillings and 6 pence, the value 
— quired. 

Independent of the mark or dot which distinguishes between 
integers and decimals, the fundamental rules — viz. Addition, Sub- 
traction, Multiplication, and Division — are in all respects the same 
as in Simple Arithmetic ; and an example in each, illustrative of 
placing the separating point, will no doubt render the whole system 
sufficiently intelligible, even to the dullest capacity. 



2^^ 



Decimal ARirnMETic. 



Ex, 1. Add into one sum the following integers and decimals: 

16-625; 11-4; 20'7881 ; 12-125; 8-04; and 7 '002. 

16-625 
11-4: 
20-7831 
12125 

8-04 
7-002 

75*9751 = the sum required. 

Subtract 119*80764 from 234'98276. 

234-98276 
119-80764 



Ex. 2. 



Ex. 



115-17512 = the remainder required. 



Multiply 62-10372 by 16-732. 

62-10372 
16-732 

12420744 
18631116 
43472604 
87262232 
6210372 



10391 1944304 — the product required. 

Observe that the number of figures in the product from the 
right hand, accounted as decimals, are equal to the number of 
decimals in the multiplier and nmltipiicand taken together. 

Ex. 4. Divide 39*375 by 9'25. 

9'2o) 39-375 (4-256 — the quotient required. 
3700 



2375 
1850 

6250 
4625 

6-250 
5550 

700 



Observe that the nunAer of deci- 
mals, in the divisor and quotient 
together, must be equal to the 
number in the dividend. 



.N<Wf«— *rhe operation ml?ht be still continued, so as to reduce the quotient toJfc » 
degree of greater exaciiUtdi.; but in practice Jt is quite unnecessary, being even now 
reduced to a measure of greater nicety than is commonly required. 



MKNHIillATK^N. 



21 



MKNSl J RATION. 

in^^llHliriliioil in l.lw^ iix'MmmI of i'n\v\i\a\.\nj,r |,|m' <'oinj)iinitiv<5 
iriAji^iii'lSiVlrA At fi^iir<'H, n,n<l il- jh (livi<l('«l into ( w«» jmrl«—jHcnHU ration 
of Su|)<!rrK'i<'H or Siii-fiuTH, immI MciiHiiinl i«Mi of I*"<»IiMh. 

Tlu' mn>^iiilii<lo of a t^ylvi'fii'^1 il4 C'llll<5<l iln nnii, himI Ih IIh- Hpiu-o 
iiH*,loH(Ml Iki1,w<i<'Ii il'H l>ouii<lHry linoH. 

TIk^ iii/ijL^iiiliKlo of ji Ixxly Ik cuIIcmI itn Holi<l <onl,<iilH, uikI Ik 
4jxj>rcHH<;(l ill ('-ubi<' f<'ol,, iiu'luiw; <<!;-<;. 

jTlf'ii^iiritlioii oV Nii|K*i*Vi«'i<'N. 



Hqniirn. 

H A 



llcclanulo. 



Ji , -. 



A StiUAKi'; Ik ii (iiuulriljitoi/il li^un-, \vlii<li Iuih all IIh h1<]<h <'<jiiu1, 
aii<i (ill it'H iuigl<;H right luiglcH. 

A KK<:TAN<ii,K in II four Hi'dtrJ i'lu^Mvv, wlii<li liiiH jIh unj^icH, rij^lit 
an^loH, aiid ilK ojH)oh)I,<5 hIjIch [mi'iulyl. 




^ C 



A ItifOMiMiH in <i parallolo^n'firri, vvIiom*- vidcs j 
an^l<^H ar<i iiol, ri^lit aii^lcH. 

A RiioMiioii) IK ii piirallclo^raiii, wlioM<ui<ljac<' 
rikI wliokd anp^l<!rt ai'<; not, v\^\\\. aiij/;l<'H. 
• A TuAPKZoii* is a four Hi(l<Ml W^nin', wlilrh Im 
parullul. 

■ ' A (Unci.i'. i.H u figuro houtKJcd l>y one Jitic, 
f<n*<in('<», find Irt Hudi l.Iuil- idl liiu-rt <lr/ivvri to tlio 
oir«utiif(rrorM'() froni a d^iriain jxdril, willilii tlin 
'fljj;uh», called Mut (Wtitro, aro (Mjiial to <'ji<*li 
oilior. Any of IIwhc TincH in failed a radhiM; 
and a lino drawn (lirou^li I|h> eenire, 1<;rini- 
naf-in)LC holli wayn in tim eJn'UFnfirencn, in 
(tailed a diafHeicr. 'I'lie j»ortion of ciiclo cut 
off l»y a dianutti-r in ('nll<'<| a Monii'irele. 

An Aitc of a <'ir<'lo in any j»oiti<>ii <»f tlio rir<-ur 



— J- 



o (.(j. 

t hid. 

hut t 
mile 



»al, hut whoMO 



•;i aie I 
wo <»f 
d tip 



ne.tjinil, 

I 
ilH HideH 

eireiuri- 

A 



28 MeNSI RATIO X. 



A Segment of a circle is a figure contained by an arc and its 
chord. 

A Versed Sine is a line drawn from the middle of a chord per- 
pendicular to the circumference. 

A Sector of a circle is a figure contained by two radii and an 
arc, as A C B E. 

PROBLEM I. 

To fill d the area of any parallelogram. 

Rule. Multiply the length by the perpendicular height, and the 
product will be the area. 

Example. Required the area of a rhomboid whose length AB=: 
20*5, and perpendicular height aA = ll'^S. 

20 5 X 11.75 = 240-875, the area. 

Note— In a square, or rectangle, the perpendicular height is the breadth : therefore, to 
find the areas of a square and rectangle, multiply the length by the breadth, 

PROBLEM IL 
To find the area of a trapezoid. 
Rule, Add together the two parallel sides, multiply their sum 
by the breadth or height, and half the product is the area. 

Example. Required the area of a trapezoid whose sides A B and 
CD are 145 and 10'25, and breadth, a A = 7*25. 

B A 



14-5 + 10-25 X 7-25 

r = 89-71875, 

the area. 



PROBLEM IIL 

To find the area of a triangle. 

Rule, Multiply one of its sides as a base by a perpendicular let 
fall from the opposite angle, and take half the product for the 
area. 

Or, from half the sum of the three sides subtract each side sepa- 
rately, and multiply the three remainders so obtained and the half 
sum together, and the square root of the product will be the area. 

Example 1. Required the area of a triangle ABC, whose base 
AB = 16-5, and perpendicular D C = 10-25. 

C 

16-5 X 10-26 ^^ ^^^^ 
• — = 84-5625, 

\ the area. 



Mensueatjon. 



A9 



Example 2. "What is the area of that triangle whose three sides 
are 8, 12, and 16 respectively? 



+ 12 +16 

_______ 



=r 18, the half sum of the sides; 



then, 



18 18 18 
8 12 13 



10 6 2 and |/ 18 x 10 x 6 x 2=46 •47, the area. 



PEOBLEM lY. 

If any two sides of a right-angled triangle he given, the third side may 
he found hy the following rules. 

1. — To the square of the base add the square of the perpen 
dicular ; and the square root of the sum will be the hypothenuse or 
longest side. 

2. — Multiply the sum of the hypothenuse, and one side by their 
difference ; and the square root of the product will be the other 
side. 

Example 1. Given the base A B =16, and perpendicular B C = 
12; required the length of the hypothenuse A C. 





VW + 12"^ = 20, the length of the hypo- 
thenuse A C 



Example 2. Given the base A B = 16, and hypothenuse A C =: 20 ; 
required the length of the perpendicular B C. 




'V^20 + 16 x 4 = 12, length of the perpendicular B C. 

Note.—'Yhe. diagonal line, or hypothenuse in a square, is equal to ihe square root of ; 
twice the square of the side. And the side of a square is equal to the square root of i 
half Ihe square of its diagonal. | 

Thus suppose each side of a square equal 12 feet : 

12^^ X 2 = V 288 = 169705 feet, the diagonal. Or, | 

~ ■ = Vl44 = 12 feet, the length of each side. 

Similar triangles, or those which are 
equi-angular to each other, have the 
sides about tlieir equal angles propor- 
tional ; tlius, in the annexed figure the 
triangles ABC and CDE are similar, 
and therefore have tlie sides about the 
equal angles proportional : 



3* 




80 



Mensuration. 



A C : B C : : 
AB:BC: 



CE: 
CD: 



DE; 
B E, (fee. 



The utility, tlien, of the above triangles for practical purposes, 
as, for instance, ascertaining the heights of buildings, <fec., will be 
seen from the following : 

Suppose D E to be an eminence, of which it is required to find 
the height, and E C the length of the shadow cast by the sun ; then, 
in order to find D E, we may erect perpendicularly at C a pole of 
any known length, as B C, and after measuring the length of its 
shadow A C, state — as the length of the pole's shadow is to the 
height of the pole itself, so is the length of the shadow of D E to 
the height of D E ; or. 

As AC: CB: : CE: ED; 

and supposinir A C = 6 feet, B C = 4 feet, and C E = 30 feet, then 
E D would be 20 feet. 

Again, supposing we wished to find the dis- 
tance betw^een two objects A and B ; draw D B 
of any length at right angles to A B, and in D B 
take any point C, through which draw A E ; also, 
at D, at right angles to D B, draw D E, making 
the triangle D E C, and state, 

As D C : D E : : B C : B A. 



PROBLEM V. 

To find the area of any regular polygon. 

Rule. Multiply the sum of its sides by a perpendicular drawn 
from its centre to one of its sides, and take half the product for the 
area. 

Or, multiply the square of the side of a pol3'gon (from three to 
tw^elve sides) by the numbers in the fourth column of the table for 
polygons, opposite the number of sides required, and the product 
X' will be the area nearly. 

Example 1. Required the area of the regular 
pentagon ABODE, each side being '7*5, and 
perpendicular F O r= 6'4. 

7'o X 5 X 6'4 

= 120, the area. 

B^^ — ' — :k ^ 

Example 2. "What is the area of a regular hexagon, each side 
being 8*75 in length ? 

8-75^ X 2-598 — 199-009875, t\\e area. 





Mensuration, 



31 



Table o/^ 


multipliers for polyg 


0118 from three to twelve sides. 


Names. 


Sides. 


Multipliers. 


Multipliers. 


Multipliers. 


Areas. 


Trigon 


3 


2 


1-73 


•579 


-433 


Tetragon . . . 


4 


1-41 


1-412 


-705 


1-000 


Pentagon. . . 


5 


1-238 


1-174 


-852 


1-72 


Hexagon . . . 


6 


1-156 


r= Radius. 


= Length 
of side. 


2-598 


Heptagon . . 


7 


1-11 


•867 


M6 


3-684 


Octagon . . . 


8 


1-08 


•765 


1-307 


4-828 


Nonagon. . . 


9 


1-062 


-681 


1-47 


6-1818 


Decagon 


10 


1-05 


-616 


1-625 


7-694 


Undecagon . 


11 


1-04 


-561 


1-777 


9-365 


Dodecagon. . 


12 


1-037 


•515625 


1-94 


11-196 



1. TJie breadth of a polygon given, to find the radius of a circle to 
contain that polygon. 

Rule. Multiply half the breadth of the polygon by the numbers 
in the first column opposite to its name, or number of sides, and the 
product will be the radius of a circle to contain that pol3'gon. 

And if the polygon have an unequal number of sides, the half 
breadth is accounted from its centre to one of its sides. 

2. The radius of a circle giveyiy to find the length of side. 

Rule. Multiply the radius of any circle by the numbers in the 
second column opposite the polygon required, and the product will 
be the length of side nearly that Avill divide that circle into the 
proposed number of sides. And, 

3. The length of side given, to find the radius. 

Rule. Multiply the given length of side by the numbers in the 
third column opposite the polygon required, and the product will 
be the radius of a circle to contain that polygon. 

Example 1. Required the radius of a circle 
to contain an octagon, whose breadth x\ B = ^, 
18"5 inches. 

Half of 18-5 — 9-25, and 9-25 x 1-08 = 9-99 
or ten inches nearly, the radius of the circle 
CD. 




Example 2. Given 
length of side D C. 



the radius OD=r9-99 inches, required the 
9-99 X -765 ==7-64285, the length of side. 



. 32 Mensuration. 

I 

I Example 3. Given the leogth of side D C = 7*64235 ; required the 
radius DO. 

7-64235 X 1-307 = 9-98855145, or 9-99 in. nearly. 

PROBLEM YL 

Having the dicimeter of a circle given^ to find the circumference ; 
or the circumference given, to find the diameter. 

Rule 1. As 7 is to 22, so is the diameter to the circumference. 
Or, as 22 is to 7, so is the circumference to the diameter. 
2. As 1 is to 3*1416, so is the diameter to the circumference. 
Or, as 3-1416 is to 1, so is the circumference to the diameter. 
Example 1. Required the circumference of a circle when .the 
dianieter is 23o. ''injjv ^ : 

^ — — — - — r= 73|, the circumference. 

Example 2. The circumference of a circle is 73f, required the 
diameter. 

V - ■ - /jgfi ^ Y 

— X 23-5, the diameter. 

22 

Example 3. Required the circumference of a circle whose diame- 
ter is SO. ■*'" ^ ' '■"" ^'''"- \ 
3-1416 X 30 =r 94-248, the circumference. 

Example 4. What is the diameter of a circle when the circum- 
ference is 94*248 ? 

■ 94-248 -f- 3*1416 = 30, the diameter. 

PROBLEM YIL 

To find the length of any arc of a circle. 

Rule. Subtract the chord of the whole arc from eight times the 
chord of half the arc ; and | of the remainder is the length of the 
arc nearly. 

" Example. Required the length of the arc ABC; the chord of 
half the arc AB = 19*8, and chord of the whole arc A'C =34*4.' 
B .19-8 X 8 =158-4, and 

^^•^'^~— ^ = 41-33, the length of 
3 
the arc. 

PROBLEM YIIL 

To find the diameter of a circle, hy having the chord and versed dne 
given. 

Rule. Divide the square of half the chord by the versed sine, to ! 



Mensuratfox. 38 




the quotient of which add the versed sine, and the sum will be the 
diameter. 

Or, if the sum of the squares of the semichord and versed sine be 
divided by the versed sine, the quotient will be the diameter of the 
circle to which that segment corresponds. 

Example. Given the chord AB = 24, and 
versed sine C D = 8 ; required the diameter of 
the circle C E. 

Half the chord = 12, and 12^-v-8:=:18 + 8 = 
26, the diameter. 

1 2^ + 8^^ 
Or, — = 26, as before. 



PROBLEM IX. 

To find the area of an ellipsis, or oval. \ 

Rule. Multiply the longest diameter by the shortest, and the : 



product by '7854; the result is the area. 

An oval is 25 inches by 16*5 : what are its superficial contents ? \ 

2o X 16-0 — 412-5 X '7854 = 323-9775 inches, the area. | 

i 

Ao?e.— Multiply half the sum of the two diameters by 31416, and the product is the i 

circumference of the oval or ellipsis. I 



PROBLEM X. 

To find the area of a parabola, or its segment. 

Rule. Multiply the base by the perpendicular height, and two- 
thirds of the product is the area. 

^ What is the area of a parabola whose base is 20 feet and height 
12? 

240 X 2 
20 X 12 = • = 160 feet, the area. 



Sotne of the properties of a circle. 

1. It is the most capacious of all plane figures, or contains the 
greatest area within the same perimeter or outline. 

2. The areas of circles are to each other as the squares of their 
diameters, or of their radii. 

3. Any circle whose diameter is double that of another, con- 
tains four times the area of the other. 

4. The area of a circle is equal to the area of a triangle whose 
base is equal to the circumference, and perpendicular equal to the 
radius. 



34 Mensuration. 

■:l± -^— ^ ■ ■ 



I 



5. The area of a circle is equal to the rectangle of its radiu^ 
and a right line equal to half its circTimference, ' ''" '-^ ' ' ' 

6. The area of a circle is found by squaring the diameter, anc^ 
I multiplying by the decimal "7854; or by multiply i rig'- irW'citc^iimf 
i ference by the radius, and dividing the product by 2? ' ' ■''*'•'-•• ; 

Example 1. Required the area of a circle, the ^diameter bein^ 

30-5. . - ■ . -H ... ; .:-..M;^• -o' , 

30-5^ X -7854 = 730-618350, the area required. 

Example 2. What is the area of a circle when the diameter is 1 ? 
In this case the circumference is 3*1416, half of which is l*57b8, 
and half of lz=z '5; then 1*5708 x '6 = •7854^ the area. ■ 

Haviyig the area of a circle given, to find the diameter. 

Rule. As 355 is to 452, so is the area to the square of the dia- 
meter. 

Or, multiply the square root of the area by 1*12837, and the 
product will be the diameter. 

"Or, divide the area by the decimal •7854, and extract the square 
root. 

Example. Required the diameter of that circle whose area is 
122-7I875. 



|/l22-71875x452 

— = 12*5, .diameter. 



355 

Or, yi22*7l875 =: 11*077 ; and 11*077 x l']2837 = 12-49895, or 
12*5, diameter. 

PROBLEM XL 
To find the area of a sector of a circle, mf. ..tj7jI 

Rule. Multiply the length of the arc by the radius ^^f*:^ffe^rpl«^, 
arid half the product will be the area. " •'-'•^ ' ^''" ^'^ * '-^ *^ 

Example. Required the area of a sector of 
a circle whose arc A B C = 26*666, and ra- 
dius B O = 16*9. 

26*666 X 16*9 ^^^ „^^^ ^^ 

— = 225*3277, the area. 

2 y '5iu\)^ V 



PROBLEM XIL 

To find the area of a segment of a circle. 

Rule. Multiply the versed sine by the decimal '626, to the square 
of the product add the square of half the chord; multiply twice 
the square root of the sum by f of the versed sine, and the product 
#ill be the area. 




MENSURATION. 



35 




Example, Required tke area of a segment of a circle whose 
efeord A B :=^ 48, and versed sine C D = 18. 

18 X -626 - 11-268^=: 126-967824; which 
add to 576, being the square of half the 
chord = 702967824, twice the square root 
©f whick is 53-026 X 12.; being f of the 
veiled sine — 63f)'3r2, the area. 

The following is a near approximate to the preceding rule : 

To the cube of th« versed sine, divided by twice the length of 
the chord, add f of the product of the chord, multiplied by the 
versed sine ; and the sum will be the area of the segment nearly. 
Take the last example : 

Versed sine = 18, and chord 48, then, p =r eO-'Z ; and 

48 X 2 

48 X 18 X 2 

— 576 + 60^7 = 636-7, the area nearly. 

o 

Or, the area i>f a segment may be found by finding the area of a 
sector having the same radius as the segment ; then deducting the 
area of the triangle, leaves the area of the segment. 

PROBLEM XIII. 

Topnd tke area xyf a •circular ring or space included between two 
concentric circles. 

Rule. Add the inside and outside diameters together, multiply 
the sum by their difference, and by -7854, and the product will be 
the area. 

Example. The diameters of two concentric 
circles, AB and CD, are 10 and 6; required 
the area of the ring or space contained between 
them. 




10 + 6 X 4 X •7854 = 50-2656, the area. 



PROBLEM XIY. 
To find the area of an ellipsis. 
Rule. Multiply the transverse or longer diameter by the conju- 
gate or shorter diameter, and by -7854, and i 
the product will be the area. 

Example. Required the area of an ellip- 
sis whose longer diameter A B =: 1*2, and 
shorter diameter C D == 9. 

13 X 9 X '7854 = 84-8232, the area. C 

JVij/e.'-lfhalfthe sum of the two diameters be multiplied by 3 ltl6, the product will 
be the circumference of the ellipsis., 

3 1416 X 21 
Thus 12 + 9 ^ 21, and ^ = 36 1384, the circumference. 




36 



Mensuration. 



Meii§uratioii of §olid§. 

By solids are meant all bodies, whether solid, fluid, or bounded ' 
space, that can be comprehended within length, breadth, and! 
thickness. ' 

PROBLEM I. I 

I 
To find the convex surface and solid conterit of a cylinder. j 

Rule 1. Multiply the circumference of the base by the height of j 
the cylinder, and the product is the convex surface. : • | 

Rule 2. Multiply the area of the base by the height of the cylin-! 
der, and the product is the solid content. i 

Example 1. Required the convex surface of the cylinder A B C D, ! 

whose base A B = 32 inches, and perpendicular height B C = 6 

feet. 
r ^ T^ 

3-1416 X 32 X 12 inches = Y2S8-2464 square or su- ; 

perficial inches, and 7238'2464 -t- 144 = 50'2658 super- 

licial feet. 

Example 2. Required the solid content, in cubic 

inches and cubic feet, of the cylinder as above. 

82" X •7854 X 12 = 5*7 905 'QY 12 cubic inches, and 
57905-9712 ^ 1728 =: 33*5104 cubic feet. 

Example 3. Suppose the cylinder A B C I) be intended to contain 
a fluid, and that the sides and bottom are each one inch in thick- 
ness, how many imperial gallons would it contain ? 

32 — 2 r= 30 inches diameter; and 72 — 1=71 inches deep;! 
,, 30^^ X -7854 X71 ^^^ ,, ! 

^^"^ 277"^274 "^ ^ ' 

Or, 50187 06 X '003607 = 181, as before. 



B 



PROBLEM IL I 

To determine the diniensio3is of any cylindrical vessel^ lehereby to 

contain the greatest cubical contents, hounded by the least superficial \ 

surface. 

Rule. Multiply the givjen cubical contents by 2*56, and the cube 

root of the product equal the diameter, and half the diameter equal 

the depth. i 

Example. Suppose a C3^1indrieal vessel is to be made so as to : 

contain 600 cubic feet, and of such dimensions as to require the ; 

least possible materials by which it is constructed, what must be 

its depth and diameter ? \ 

600 X 2-56 = n/i536 = 11*5379 feet diameter, \ 

aud 11-5379 -r- 2 = 5*76895 feet in depth. 

Note.—K the vessel is to be constructed with two ends, then the cube root of four 
times' the solidity divided by 31416 equal both the length and diameter, so as to expose 
the least possible surface, or be composed of the least possible materials, of which to be 



Mfc:NSUUATION. 



'61 



PROBLEM III. 

To find the surface and solid content of a cone or pyramid. 

Rule 1. Multiply the circumference of the base by the slant 
lieight, and half the product will be the slant surface ; to which 
add the area of the base, and the product will be the whole surface. 

Rule 2. Multiply the area of the base by the perpendicular 
height, and ^ of the product will be the solid content. 

Example 1. Required the convex surface of a 
cone whose base A B = 20 inches, and slant height 
B D = 29-5. 

3-1416 X 20 X 29-5 ^^^^^^ . , 
•— — — - =: 926-772 square inches, 

A 

and divided by 144 — 6*435 superficial feet. 

ExAAfPLE 2. Required the solidity of the cone 
as above, the perpendicular C D being 28 inches. 

20^ X '7854 X 28 

= 2932-16 cubic inches, and divided by 1728 = 

o 

1-697 cubic feet. 

PROBLEM IV. 
To find the surface of the frustum of a cone or pyramid. 

Rule. Multiply the sum of the perimeters of the two ends by 
the slant height, and half the product will be the slant surface ; to 
which add the areas of the two ends, and the product will be the 
whole surface. 

Example. Required the convex surface of 
the frustum of a cone A B C D, whose base 
ABrr:20 inclics, the slant height BC=: 19, 
and top end C D :=: 11. 





3-1416 X 20 + 3-1416 x 11 x 19 
2 
= 925*2012 square inches, and divided by 144 B \^: 
== 6-425 feet nearly. 

PROBLEM V. 
To find the solid content of the f rust wn of a cone. 

Rule. To the product of the diameters of the two ends add tlie 
sum of their squares; multiply this sum by the perpendicular 
height and by -2618 ; the product is the solid content. 

Example 1. Required the solid content of the frustum in Problem 
IV., whose perpendicular EF = 18 inches. 

20 X 11 = 220, and 220 + 20^ + 11' x 18 x -2618 = 3491-8884 
cubic inches, and divided by 1728 = 2 0208 cubic feet nearly. 



88 



Mensuration. 




Example 2. Required the content, in imperial gallons, of the 
inverted frustum of a cone A B C D, whose inner dimensions are 3^ 
feet deep, 18 inches diameter at bottom, and 22 inches diameter at 
top. 

22 X 18 = 396, and 396 + 22'-^ + Ib^ X 42 
13238*7024 
X -2618 =: —————— = 47 •'745 galls, nearly. 

277*274 ^ -^ 

Or, 13238-7024 x 0*00360654 = 47-75 gal- 
lons nearly, as before. 

PROBLEM yi. 

To find the solid content of the frustum of a pyramid. 
Rule. To the sum of the areas of the two ends add the square 
root of their product ; multiply this sum by the 
perpendicular height, and -J- of the product is the 
solid content. 

Example. Required the solid content of the 
frustum of a pyramid A B C D, whose perpen- 
dicular height = 24 inches, the area of the base 
= 144 inches, and area of the top end = 64. 

144 + 64 r= 208, and Vl44 x 64 = 96 ; then 
208 + 96 X 24 



144 X 64 
=: 2432 cubic inches, and -^ 1728 




= 1*4074 cubic feet nearly. 

PROBLEM VIL 

To find the solidity of a wedge. 
Rule. To the length of the wedge add twice the length of the 
base ; multiply that sum by the height, and by the breadth of the 
base, and one-sixth of the product will be the solidity. 

Example. Required the content in cubic 

inches of the wedge ABODE, whose base 

ABC = 12 inches long and 4 inches broad, the 

length of the edge D E = 10 inches, and perpen- 

- dicular height r E = 20 inches. 



S) 



10+24 X 20 X 4 



■ 452-83 cubic inches. 



PROBLEM VIII. 

To find the convex surface and solid content of a sphere or globe. 

Rule 1. Multiply the square of the diameter by 3-1^:16 jtjie 
product will be the convex superfi<jie3. " f'^'^^^ ui^^i^ 



Mensuration. 



89 




1728 = 5-0243 cubic 



Rule 2. Multiply the cube of the diameter by -5236, and the 
product is the solid content. 

Example 1. Required the convex surface 
of a sphere, whose diameter A B = 25^ 
inches. 

25-5^ X 3-1416 =: 2042-8254 square inches, 
-i- 144 = 14*1862 square or superficial feet. 

Example 2. Required the solid content of 
a sphere whose diameter A B — 25i inches. 

25-5^ X -5236 = 8682*00795 cubic inches ; - 
feet. 

PROBLEM IX. 

To find the convex surface and solid content of the segment of a 

spjiere. 
Rule 1. Multiply the height of the segment by the whole cir- 
CHmference of the sphere, and the product is the curved surface. 

Rule 2. Add the square of the height to three times the 
square of the radius of the base ; multiply that j) 

sum by the height, and by '5236, and the pro- 
duct is the solid content. G^ 



Example 1. The diameter A B of the sphere 
ABCD = 20 inches; what is the convex sur- 
face of that segment of it whose height E D = 
8 inches? 




2 



144 r= 3-49 super- 



3-1416 X 20 X 8 = 502-656 square inches 
ficial feet. 

Example 2. The base F G of the segment F D G = 18 inches, and 
perpendicular ED — 8 ; what is the solid content ? 

8* = 64, and 9^ x 3 = 243 ; then 243+64 x 8 x -5236 = 1285-9616 
cubic inches, ~ 1728 = '7441 cubic feet. 

Example 3. Suppose A B C D to be a sugar-pan, and that the 
diameter of the mouth A B is 4 feet, the depth D C being 25 inches, 
how many imperial gallons will it contain ? 
25^ = 625, and 24^^ x 3 = 1728 ; then B 

r728 4- 625 X 25 x -6236 ^^^^^''^'^ 



111*084 gallons. 



277-274 




PROBLEM X. 

To find the solidity of a spheroid. 

Rule. Multiply the square of the revolving axis by the fixed 
axis, and by -6236, and the product will be the solidity. 



49 



Mensuration. 




Example 1. Requh'ed the Bolid content of the 
prolate spheroid A B C D, whose fixed axis A C is 
60, and revolving axis B D 30. 

SO'* X 50 X -5236 = 23562, the solidity. 

Example 2. What is the solid content of an oblate spheroid, the 
fixed axis being 30, and revolving axis 50? 

60^ X 30 X -5236 =^ 392*70, the solid content 



PROBLEM XL 

To find the solidity of the segment of a spheroid when the base is 
circular or parallel to the revolving axis. 

Rule. From triple the fixed axis take double the height of the 
segment ; multiply the difference by the square of the height, and by 
•5236 ; then say, as the square of the fixed axis is to the square of 
the revolving axis, so is the former product to the solidity. 

Example 1. Required the solid content of the segment ABC, 
whose height B r is 10; the revolving axis E F being 40, and fixed 
axis B D 25. p 

25 X 3 — 10 X 2 =: 55, and 55 x 10' ^^ 

25*^ 



•5236 = 
2879-8 



2879-8. Then, 
7372-3 nearly. 



40^ 




Example 2. What is the solid content of the segment of a spheroid 
whose height =z 20 inches, the revolving axis being 25, and fixed 
axis 50 ? 

50 X 3 -"20 x 2 = 110, and 110 x 20^ x -5236 := 23038-4; then, 
as 50^* : 25^*:: 23038-4 : 6759*6 inches, the solid content 



PROBLEM XIL 

To find the convex suiface and solid content of a cylindric ring. 
Rule ]. Multiply the thickness of the ring added to the inner 
diameter by the thickness and by 9-8698, and the product w^ill be 
the convex surface. 

Rule 2. To the thickness of the ring add the inner diameter ; 
multiply that sum by the square of the thickness and by 2-4674, and 
the product will be the solid content 

Example 1. The thickness of a cylindric 
ring A C or D B =: 2 inches, and inner diame- 
ter = 18, required the convex superficies. 

18+2 X 2 X 9-8698 = 394-792 square 
inches, and -h 144 = 2*741 superficial feet 
nearly. 




Mensuration. 



41- 



Example 2. Kequired the solid content of the ring as above. 

18 + 2 X 2^x 2-46'74 =: 197 '392 cubic inches, and -f- 1128 = '114 
cubic feet. 

Note.— A cubic foot is equal to 1728 cubic inches, 

or 2200 cylindrical incheSi 
or 33to spherical inches, 
or 6600 conjcal inches. 

Also, the cubic foot being considered unity, or 1, 

A cylinder 1 foot in diameter and 1 foot in length* •••• « = "7854 

A sphere 1 foot in diameter. = 5236 

And a cone 1 foot in diameter at the base and 1 foot in height = '2619 



]>eeiinal Approximatioiii$9 

FOR FACILITATING CALCULATIONS IN MENSURATION. 



Lineal feet multiplied by 

** yards, ** 

Square inches, " 
" yards, 

Circular inches, " 

Cylindrical inches, " 
feet. 

Cubic inches, " 
" feet, 

" inches, " 

Cylindrical feet, " 

" inches, " 

Cubic inches, " 



Cylindrical inches. 



Avoirdupois lbs., 



•00019 

•000568 

•007 

•0002067 

•00546 

•0004546 

•02909 

•00058 

•03704 

•232 

•003607 

•895 

•002832 

•263 

•281 

•283 

•3225 

•3037 

•26 

•4103 

•2636 

•4908 

•2065 

•2168 

•2223 

•2533 

•2385 

•2042 

•8223 

•207 

•3854 

•009 

•00045 



= miles. 

= square feet. 

— acres. 

= square feet. 

— cubic feet. 
=r cubic yards. 
= cubic feet. 
=z cubic 3'ards. 

= imperial gallons. 



: lbs. avs. of cast iron. 

" wrought do. 

steel. 
" copper. 

" brass. 

" zinc. 

lead. 

tin. 
" mercury. 

" cast iron. 

" wrought iron. 

steel. 
" copper. 

" brass. 

" zinc. 

" lead. 

" tin. 

*• mercury, 

cwts. 
tons. 



^% Instrumental Arithmetic. 



INSTRUMENTAL AEITHMETIC ; 

OR, UTILITY OF THE SLIDE RULE. 

The slide rule is an instrument by -which the greater portion of 
operations in arithmetic and mensuration may be advantageously 
performed, provided the lines of division and gauge points be made 
properly correct, and their several values familiarly understood. 

The lines of division are distinguished by the letters A B C D, 
A B and C being each divided alike, and containing what is termed 
a double radius, or double series of logarithmic numbers, each series 
being supposed to be divided into 1000 equal parts, and distributed 
along the radius in the following manner : 

From 1 to 2 contains 301 of those parts, being the log. of 2. 



3 


477 




i» " ° 


3. 


4 


602 




" 


4. 


5 


699 




?' 


5. 


6 


778 




•' 


6. 


7 


845 




" 


7. 


8 


903 




" 


8. 


9 


954 




♦' 


9. 




1000 


being the 


whole number. 





The line D, on the improved rules, consists of only a single 
radius; and although of larger radius, the logarithmic series is the 
same, and disposed of along the line in a similar proportion, form- 
ing exactly a line of square roots to the numbers on the lines B C. 

JVumeration. 

Numeration teaches us to estimate or properly value the num- 
bers and divisions on the rule in an arithmetical form. 

Their values are all entirely governed by the value set upon the 
first figure, and, being decimally reckoned, advance tenfold from 
the commencement to the termination of each radius : thus, suppose 
1 at the joint be one, the 1 in the middle of the rule is ten, and 1 at 
the end one hundred. Again, suppose 1 at the joint ten, 1 in the 
middle is 3 00, and 1 or 10 at the end is 1000, <fec., the intermediate 
divisions on which complete the whoie system of its notation. 

To Multiply ]¥iimbers by tlie Rule. 

Set 1 on B opposite to the multiplier on A; and against the 
number to be multiplied on B is the product on A. 
Multiply 6 by 4. 
Set 1 on B to 4 on A: and against 6 on B is 24 on A. The slide thus set, against 

7 on B is 28 on A 

8 " 32 

9 " 36 " 

10 " 40 " 

11 " 44 

12 " 48 " 
15 " 60 •* 

25 " 100, dec, &C 



Instrumental Arithmetic. 43 



To divide TVuinbers upon the Rule. 

Set the divisor ou B to 1 on A, aad against the number to be 
divided on B is the quotient on A. 
Divide 63 by 3. 

Set 3 on B to I on A, and against 63 on B is 21 on A. 

Proportion, or Rule of Tliree I>irect. 

Rule, Set the first term on B to the second on A. and against the 
third upon B is the fourth upon A. 

1. If 4 yards of cloth cost 38 shillings, what will 30 3'ards cost at 
the same rate ? 

Set 4 on B to 38 on A, and against 30 on B is 285 shillings on A. 

2. Suppose I pay 31s. 6 J. for 3 cwt. of iron, at what rate is that 
per ton ? I ton = 20 cwt. ^ 

Set 3 upon B to 315 upon A, and against 20 upon B is 219 upon A. 

Rule of Tliree Inverse. 

Ride. Invert the slide, and the operation is the same as direct 
proportion. 

1. I know that six men are capable of performing a certain given 
portion of work in eight days, but I want the same performed in 
three : how many men must there be employed ? 

Set 6 upon C to 8 upon A, and against 3 upon C is 16 upon A. 

2. The lever of a safety valve is 20 inches in length, and 5 inches 
between the fixed end and centre of the valve: what weight must 
there be placed on the end of the lever to equipoise a force or pres- 
sure of 40 lbs. tending to raise the valve ? 

Set 5 upon C to 40 upon A, and against 20 on C is 10 on A. 

3. If 8 J yards of cloth, \^ yards in width, be a sufiicient quantity, 
how much will be required of that which is only |ths in width, to 
effect the same purpose ? 

Set rs on C to 8 75 on A, and against 875 upon C is 15 yards upon A. 

Square and Cube Roots of TVunibers. 

On the engineer's rule, when the lines C and D are equal at both 
ends, C is a table of squares, and D a table of roots, as — 

Squares, 1 4 9 16 25 36 49 64 81 on C. 
Roots. 12 3 4 5 6 7 8 9 on I). 

To find the geometrical mean proportion hetioeen two numhers. 

Set one of the numbers upon C to the same number upon D, and 
against the other number upon C is the mean number or side of an 
equal square upon D. 

Required the mean proportion between 20 and 45. 

Set 20 upon C to 20 upon D, and against 45 upon C is SO on D. 



44 Instrumental Arithmetic. 



To cube any number, set the number upon C to 1 or 10 upon D, 
and against the same number upon D is the cube number upon C. 
Required the cube of 4. 

Set 4 upon C to 1 or 10 upon D, and against 4 upon D is 64 upon C. 

To extract the cube root of any number, invert the slide, and set 
the number upon B to 1 or 10 upon D, and where two numbers of 
equal value coincide, on the lines B D, is the root of the given 
number. 

Required the cube root of 64. 

Set 64 upon B to 1 or 10 upon D, and against 4 upon B is 64 upon D, or root of the given 
number. 

On the common rule, when 1 in the middle of the line C is set 
opposite to 10 on D, then C is a table of squares, and D a table of 
roots. 

To cube any number by this rule, set the number upon C to 10 
upon D, and against the same number upon D is the cube upon C. 

Mensuration of Surface. 

1. Squares, Rectangles, &c. 

Rule. When the length is given in feet, and tlie breadth in 
inches, set the breadth on B to 12 on A; and against the length on 
A is the content in square feet on B. 

If the dimensions are all inches, set the breadth on B to 144 upon 
A ; and against the length upon A is the number of square feet on B. 

Required the content of a board 15 inches broad and 14 feet long. 
Set 15 upon B to 12 upon A ; and against 14 upon A is 17 5 square feet on B. 

2. Circles, Polygons, <hc. 

Rule. Set '7854 upon C to 1 or 10 upon D • then will the lines 
C and D be a table of areas and diameters. 

Areas, 314 7 06 ISSe 19 63 2827 3848 5026 6361 upon C. 

Diameters, 234567 8 9 upon D. 

In the common rule, set -7854 on C to 10 on D ; then C is a line 
or table of areas, and D of diameters, as before. 

Set 1 upon B to 22 upon A ; then B and A form or become a 
table of diameters and circumferences of circles. 

Circumferences, 314 6 28 9'42 1256 15 7 1885 22 2513 2827 upon A. 
Diameters, 123 4 56 789 upon B. 

Polygons frmn 3 ^o 12 sides. Set the gauge-point upon C to 1 or 
10 upon D; and against the length of one side upon D is the area 
upon C. 

Sides, 3 5 6 7 8 9 10 11 12. 

Gauge-points, 433 17 2-6 3 63 482 618 7 69 937 1117. 

Required the area of an equilateral triangle, each side 12 incTies 
in length. 

Set 433 upon G to 1 upon D ; and against 12 upon D are 625 square inches upon C. 



??^ffl^p 




Instrumental 


Arithmetic. 






45 


TABLE OF GAUGE-POINTS FOR THE ENGINEER'S RULE. 




Names. 


F, F, F. F. I, I. 


I, I, I. 


F, 1. 


I.I 


F. 


1. 


Cubic inches, . . 


578 


83 


1728 


106 


1273 


105 


121 


Cubic feet, , 






1 


144 


1 


1833 


22 


121 


33 


Imperial gallons 






163 


231 


277 


294 


353 


306 


529 


Water in lbs., 






16 


23 


276 


293 


352 


305 


528 


Gold 




. 


814 1175 


141 


149 


178 


155 


269 


Silver " 




'. j 15 


216 


261 


276 


334 


! 286 


5 


Mercury " 




. ! 118 


169 


203 


216 


258 


! 225 


389 


Brass 






193 


177 


333 


354 


424 


369 


637 


Copper " 






18 


26 


319 


331 


397 


i 345 


596 


Lead 






141 


203 


243 


258 


31 


: 27 


465 


Wro't iron " 






207 


297 


357 


338 


453 


' 394 


682 


Cast " " 






222 


32 


384 


407 


489 


; 424 


733^ 


Tin 






219 


315 


378 


401 


481 


419 


728 


Steel " 






202 


292 


352 


372 


448 


385 


671 


Coal 






127 


183 


22 


33 


28 


242 


42 


Marble 






591 


85 


102 


116 


13 


113 


195 


Freestone " 


— 


— 


632 


915 


11 


1162 


14 


141 


21 


FOR THE COMMON 


SLIDE 


RULE. 








Names. 


F, F, F 


F, I. I. 
518 


1,1, I 


F, I. 


I, I. 


F. I. 


Cubic inches, . . 


36 


624 


660 


799 


625 


113 


Cubic feet, . . 






625 


9 


108 


114 


138 


119 


206 


Water in lbs., . 






10 


144 


174 


184 


22 


191 


329 


Gold " . 






507 


735 


88 


96 


118 


939 


180 


Silver " . 






938 


136 


157 


173 


208 


173 


354 


Mercury " 






738 


122 


127 


132 


162 


141 


242 


Brass " 






12 


174 


207 


221 


265 


23 


397 


Copper " 






112 


163 


196 


207 


247 


214 


371 


Lead " . 






880 


126 


152 


162 


194 


169 


289 


Wro't iron " . 






129 


186 


222 


235 


283 


247 


423 


Cast " " . 






139 


2 


241 


254 


304 


265 


458 


Tin « . 






137 


135 


235 


25 


300 


261 


454 


Steel " . 






136 


183 


22 


233 


278 


239 


418 


Coal 






795 


114 


138 


146 


176 


151 


262 


Marble " . 






370 


53 


637 


725 


81 


72 


121 


Freestone " 






394 


57 


69 


728 


873 


755 


132 





46 TXSTRUMENTAL ARITHMETIC. 



Meii§ui*atiosi of Solidity and Capacity. 

General Rule, Set the length upon B to the gauge-point upon 
A ; and against the side of the square, or diameter on D, are the 
cubic contents, or weight in lbs. on C. 

1. Required the cubic contents of a tree 80 feet in length, and 10 
inclies quarter girt. 

Set 20 upon B to 144 (the gauge-point) upon A; and against 10 upon D is 2075 feet 
upon C. 

2. In a cylinder 9 inches in length and 1 inches diameter, how 
many cubic inches ? 

Set 9 upon B to 1273 (the gauge-point) upon A ; and against 7 on D is 346 inches on C. 
8. What is the weight of a bar of cast iron 3 inches square, and 6 
feet long ? 
Set 6 upon B to 32 (the gauge-point) upon A; and against 3 upon D is 168 lbs. upon C. 
By the common rule. 

4:. Required the weight of a cylinder of wrought iron 10 inches 
long, and 5|- diameter. j 

Set 10 upon B to 233 (the gauge-point) upon A ; and against 5}i upon D is 6665 lbs. ' 
on C. j 

5. What is the weight of a dry rope 25 yards long, and 4 inches 
circumference 'i 

Set 25 upon B to 47 (the gauge-point) upon A ; and against 4 on D is 5316 lbs. on C. 

6. What is the weight of a short linked chain 30 yards in length, 
and Y^t\\?> of an inch in diameter ? 

Set 30 upon B to 52 (the gauge-point) upon A ; and against 6 on D is 129'5 lbs. on C. 

l^aud Surveying^. 

If the dimensions taken are in chains, the gauge-point is 1 or 10; 
if in perches, 160 ; and if in yards, 4840. 

Rule. Set the length upon B to the gauge-point on A; and; 
against the breadth upon A is the content in acres upon B. i 

L Required the number of acres or contents of a field 20 chains ' 
50 links in length, and 4 chains 40 links in breadth. 
Set 20 5 on B to 1 on A ; and against 4 4 on A is 9 acres on B. 

2. In a piece of ground 440 yards long, and 44 broad, how many 
acres ? 
Set 440 upon B to 4340 on A ; and against 44 on A is 4 acres on B. 

Power of Steam-Engines. 

Condennng Engines — Rule. Set 3*5 on C to 10 on D ; then D is 
a line of diameters for cylinders, and C the corresponding number \ 
of horse power ; thus, 

Horse power, . 3K 4 5 6 8 10 12 16 20 25 30 40 50 on C. 
CD 10 in. \QX 12 13M 15K 17 18^ 21>^ 24 26^ 29K 33^ 37^^ on D. 



The Laws ok Motion, 4T 



The same is effected on the common rule by setting 5 on C to 12 
onD. 

Non-condenmig Engines. — Rule. Set the pressure of steam in lbs. 
per square inch on JB to 4 upon A; and against the cylinder's 
diameter on D is the number of horse power upon C. 

Required the power of an engine, Avhen the cylinder is 20 inches 
diameter, and steam 30 lbs. per square inch. 

Set 30 on B to 4 on A ; and against 20 on D is 30 horse power on C. 

The same is effected on the common rule by setting the force of 
the steam on B to 250 on A. 

Of Kiigine Boiler§. 

How many superficial feet are contained in a boiler 23 feet in 
length and h\ in width ? 

Set 1 upon B to 23 upon A ; and against 55 upon B is 126-5 square feet upon A. 

If 5 square feet of boiler surface be sufficient for each horse 
power, how many horse power of engine is the boiler equal to ? 
Set 5 upon B to 126 5 upon A ; and against 1 upon B is 255 upon A. 

The r.a\¥§ of Motion. 

If 3/" = mass of a material body, 
And ir= the weight of it. 

.*. W—M X 32-19; 

Or the mass of a body is equal to its weight divided by 32*19. 
Example. Find the weight of a body whose mass is 3 J : 

.-. Wr=^ 3-5 X 32-19 - 112-66 lbs. 

The gravity of a material body is its weight. Falling bodies f[dl 
through the same space in the same time, whatever may be their 
weight. A body one ton will fall to the ground no faster than a 
body one pound. 

The velocity of a body is the number of feet passed over in one 
second. 

Put V = the velocity of a falling body, at the end of t seconds , 

.-. -y = 32-19 X t. 

The quantity 32 19 is the velocity of a falling body at the end of 
one second. 

Rule^ to find the Velocity of a Falling Body at the end of any 
Number of Seconds. 

Multiply the number of seconds by 32'19, the product will be the 
velocity. 



TiJE Laws ok Motion. 



I Example. Find the velocity of a body falling from a height in 
; nine seconds : 

Velocity = 32-19 x 9 = 289-71. 

Put s for the number of feet a falling body falls through in t 
seconds : 

32-19 ^^ 



Hide to find the Space passed over hy a Falling Body in any Number 
of Seconds. 

Square the number of seconds, and multiply the result by 16*09, 
the product will be the distance passed over in feet. 

Example. A stone fell from the top of a chimney to the bottom 
in four seconds ; find the height of the chimney: 

Height of chimney = 16 09 x 16 == 2o1-U feet. 

v'' 
s z= , w^here v is the velocity. 

Bule to find the Space passed over by a FoMing Body wheti the 
Velocity is given. 

Square the velocity, and divide by 64-39 ; the quotient will be 
the number of feet passed over. 

The quantity 32*19 is frequently called the accelerating force of 
gravity, and is denoted by/. The following formulae include all 
cases that can occur in falling bodies. 

fe tv v^ 
s z= space passea over = -— ■——- = -—-; 
2 2 2/ 

2 5 / 

V r= velocity at the end of {t) seconds =/^=— - = a/ 2/s ; 

V 2 s /o"^ 

t =: time = -;-==:— r= ^ /£?; 



V v'^ _ 2s 

*^~T"~2^~'F' 

The above formulte and rules are applicable only to the case 
when the body is acted upon by the force of gravity. 

Rules and Formulce when a body is acted on by any force. 
Put M=^ mass acted on by a force of impounds. 

a = velocity at the end of a second, which is called acce- 
lerating force. 
5 = space passed over in {t) seconds, producing a velo- 
city {v). 



The Laws of Motion-, 



^ 



F 
M 


- 


V 

7"' 


Fe 




M^P' 


M 


-- 


F 



And 2^ 



Rule for finding the accelerating force of a body. 

Divide the force by the mass (remembering that mass is equal to 
weight divided by 82 19) or the velocity by the time, either 
quotient will give the accelerating force. 

Example. A force of 25 lbs. acts on a body whose weight is 84 lbs. 
Find the accelerating force. 

84 

82-19 



The mass ; 



= 2*6 nearly; 



25 



2-6 



■ = 9-62 nearly. 



The velocity at the end of 10 seconds = 9*62 x 10 — 96*2. 

Time of a Body falling down an Inclined Plane. 

Let ^ 5 C be an inclined plane, B O per- 
pendicular, and A B parallel to the horizon. 

The velocity at A in falling down A C \?> 
the same as it would be in falling perpen- 
dicularly down the height B C. 

Put t = time in falling from C to A. 

I z=z A G the length of the inclined plane. 
hz=i B C the height of ditto. 




t- 



v/ 



2f 



Let A D E B be a circle whose diameter A B is 
perpendicular to the horizon. The times of a 
body falling down any chords A D, ^ ^are equal, 
and equal to the time in falling vertically through 
AB, 

TJte Time of Oscillation of a Siiuple Pendulum, 
Let A B the length of the pendulum — /, 
And TT =r 3-1415^9, cfec. ; g — 32*19 
T = time in seconds oscillating from the point 
BioD. 

The arc -fi (7 = C I) h small. 




■■■— (^y 




^ The Laws of Motiox. 



Hide to find the Time of one Owillaiion of a Simple Pendulum. 

Divide the length of the pendulum by 32'19; extract the square 
root of this quotient, and multiply the result by 3*1416, and the pro- 
duct will be the time of oscillation in seconds. 

If L be the length of a pendulum which oscillates in one second, 



H^y 




The value of L for the latitude of London is 39*1386 inches. A 
pendulum 9|i, 4f|, 2f^^ inches long, will oscillate in a lialf, a 
third, a quarter seconds respectively. 

If 7i- be the number of oscillations made by a pendulum in one 
hour, then 

I = 3600'^ X -^ 

The time of oscillation is not dependent on the weight of the bob. 

Centrifugal Force. 

Let the weight W, placed at B, be connected 
with a cord, or wire, with the fixed point A 
1^ round which it revolves with a uniform ve- 
locity. 
Put V = velocity of rotation. 

r = A B, the length of the cord in feet. 
F = centrifugal force, or the force which is exerted to break the 
cord in the direction of its length. 

32-19 X r ' 
If n be the number of revolutions in one minute, 

.\F= — -^ X Wrn''. 

1000000 

If Whe measured in tons, then i^will be in tons also. 
If 10 be the angular velocity, 

^ Wrvr 

.'.F= — ' 

9 
If ^be the time of the weight making a complete revolution, 

,'. w — angular velocity = — = — -. 



If there be several bodies at B, 0, B, and re- 
volving round the axis passing through A, and 
perpendicular to the plane A D B Cy 




The Laws of Motion. 51 






W' + W + r'' W^ + <fcc. 



1 /. 2 

.-. F— — -- ^ 8 X 80 + 12 X 90 (. = 30178 lbs. 



Where w = angular velocity, W^ ; TF^ ; Tl^^, <fec. : the weights at 
B CD, &c., and r\ r^ r^ ifec , the distances AB, A C\ A Z>, <fec. 

Example. Let the weights at B and C be 80 and 90 lbs. respec- 
tively, revolving at a distance A B ^=^ S feet, A C =^ 1'2> feet, with a 
velocity naaking 40 revolutions per minute. Find the centrifugal 
force, or the pressure on the axis passing through A. 

^ ~ ~60 " "" T' 

16 TT^ ( ) 

— — J 8 X 80 + 12 X 90 (. 
The moment of inertia. 

If ( W, +W2 + IF3 + Ac.) P = W,ri+ W2 r/ + IFg r./ + &^. 

Each side of this equation is called the moment of inertia, and 
the distance k is called the radius of gj-ratioii of the revolving 
S3'stem. 

Let a constant force F act at a distance Af=a from the axis 
of motion. 

The angular velocity at the end of a second 

_ ff ^'^ 

The angular velocity at tlie end of one revolution 

_ 2 v]rF'~^ 



V IFi + W.J + IK3 + tfec. X k 
If a point be determined from the equation 

where G is the centre of gravity of the system, then is called the 
centre of oscillation. 

27ie values of k ht, Geometrical Solids. 
A rectangular parallelopipedon revolving about an axis passing 
through its centre of gravity, and parallel to either of its edges. 

12 ' 
wjiere b c are tlie length and breadth at right angles to the axis of 
i-e volution. 

An upright triangular prism about a vertical axis passing through 
its centre of gravity. 

2 2 

48 36 



52 * The Laws of Motion, 



The section of the prism perpendicular to the revolving axis is i 
an isosceles triangle, the base being denoted by (a), and the per- 1 
pendicular upon it from the angle contained by the equal sides j 

by (4 . . i 

In a cylinder, whose radius is (r), revolving about its axis, 






In a hollow cylinder, whose internal and external radii are a and 
b respectively, revolving about its axis, 



In a cylinder, whose radius is r and length /, revolving round a 
line at right angles to its axis, and passing through its middle, 

12 ^ 4 

In a sphere, whose radius is r, revolving about its diameter, 

2 r^ 
5 

In a hollow sphere, whose internal and external radii are (a) and 
(b) respectivel}?^, revolving about its diameter, 

_ 2ib^-a^) 
~.6(b'-ay 

In a cone, whose base is a circle, radius r, 

10 

In a cone, whose radius of base is r and height h, revolving about 
a line at right angles to its axis, and passing through its centre of 
gravity, 

80 

The square of the radius of gyration about any line in a revolv- 
ing system, is equal to the square of the radius of gyration about a 
line parallel to it passing through the centre of gravity and the | 
square of the distance from the centre of gravity to the line about i 
which the system revolves. i 



The Centre ok Percussion. 63 




Let G be the centre of gravity of any body ; 
draw A B any line about which the system re- 
volves. Let b ]) be parallel to A By and draw 
G H perpendicular to A B. 
Let K = radius of gyration when revolving 
about A B. 
]c = radius of gyration when revolving 

about Cli. 
.-. K^ = /c''+G H\ 
This important theorem will readily extend the theorems which 
are given above to most practical cases. 

The Centre of Oyration 

is that part of a body revolvhig about an axis, into tvhich, if the, 
wJtole qnantiiy of matter were collected^ the same moving force looutct 
generate the same angidar velocity. 

To find the centre of Gyration, multiply the weight of the several 
particles by the squares of their distances from the centre of 
motion, and divide the sum of the products by the weight of the 
whole mass ; the square root of the quotient will be the distance 
of the centre of gyration, from the centre of motion. 

The distances of the centre of gyration from the centre of 
motion, of different revolving bodies, are as follows: 

In a straight rod revolving about one end, the length x 'SY'TS. 

In a circular plate, revolving on its centre, the radius x '707 1. 

In a circular plate, revolving about one diameter, the radius x '5. 

In a thin circular ring, revolving about one diameter, radius 
X -7071. 

In a solid sphere, revolving about one diameter, the radius 
X -6325. 

In a thin hollow sphere, revolving about one diameter, the radius 
X -8164. 

In a cone, revolving about its axis, the radius of the base x 
•5477. 

In a right-angled cone, revolving about its vertex, the height 
X -806. 

In a paraboloid, revolving about its axis, the radius of the base 
X -5773. 

The Centre of Percussion 

is that point in a body r evolving about a fixed axis, into ivJdch the 
whole of the force or motion is collected. 

It is, therefore, that point of a revolving body which would 
stuike any obstacle with the greatest effect; and, from this pro- 
perty, it has received the name of the centre of percussion. 

_ 



A4 On Work. 



I 



The centres of oscillation and percussion are in the same point. | 

If a heavy straight bar, of uniform density, be suspended at one ! 
extremity, the distance of its centre of percussion is two-thirds of; 
its length. 

In a long slender rod of a cylindrical or prismatic shape, the : 
centre of percussion is nearly two-thirds of the length from the axisj 
of suspension. I 

In an isosceles triangle, suspended by its apex, the distance of; 
the centre of percussion is three-fourths of its altitude. In a line ori 
rod whose density varies as the distance from the point of sus.pen- ! 
sion, also in a fly-wheel, and in wheels in general, the centre of 
percussion is distant from the centre of suspension three-fourths of 
the length. 

In a very slender cone or pyramid, vibrating about its apex, the 
distance of its centre of percussion is nearly four-fifths of its length. 

On Work. 

A unit of work is one pound avoirdupois raised vertically one | 
foot. I 

If U denotes the units of work in raising W lbs. h feet — 

.-. U=: Wh, 

i 
Rule to find the ITfiits of Work in Raising a given Weight a given 

Height. 

Multiply the height in feet by the weight in pounds, the product 
will be the units of work done. 

Example. Find the units of work in raising half a ton 30 feet 

.-. U:= 1120 X 30 rr: 33600 units of work. 

It is important to observe, in the application of the above for- 
mula to practical cases, that the height (//) is the vertical distance 
through which the centre of gravity of the body whose weight is 
( W) is raised. 

Example. Find the units of work in lowering the surface of 
water in a well one yard ; the depth to the surface of water being 
40, and diameter 3 feet 

The weight of a cubic foot of water is 62^^ lbs. 

The weight of water == 9 x '7854 x 3 x 62-5 = 1325*36 lbs. 

The height through which the centre of gravity is raised = 41*5 
feet. 

.-. U= 1325-36 X41-5 = 55002 units of work. 

The work done in raising a body up an inclined plane, or any 
curved surface, is equal to the work done in raising the body ver 
tically through the height of the inclined plane. 

There are 29000 units of work done in sawing a square foot of 
green oak. 



Ox Work. 



Home Power. 

A horse power is 33000 units of work done in one minute. 
Put //, equal to the horse power, and U^ the units of work done, 
in jTliours: 

U 
.-.38000^= ——-. 
60 r 

The following results are taken from Morin : 
A Man laboring Eight Hours per Day will perform the following 
Units of Work. 

Raising his own body, 4250 

Drawing, or pushing horizontally, .... 

Pushing and drawing alternately in a vertical direction, 
Turning a handle, ....... 

Working with his arms and legs, as in rowing, 

A Man laboring Six Hours per Day. 

Raising material with a pulley, 1560 

Raising material with the hands, 1470 

Raising material upon the back, and returning empt}', . . 1126 

A Man laboring Ten Hours per Day. 
Raising material with a wheelbarrow on ramps, . . . 720 
Throwing earth to the height of five feet, , ., . , 470 

Useful Work of a Man raising Water — Duration of Labor, Eight 
Hours per Day. 
With a windlass from deep wells, . . . . . 2660 

With an upright chain pump^ 
With a Chinese wheel, ..... 
With an Archimedean screw. 
Raising water from a well with a pail and rope, 



3120 

2380 
2600 
400(J 



1730 
2167 
1505 
1054 



Work of AnhnaU, 

A horse, in a common pumping engine, .... 17550 

A mule, ditto, 11700 

An ass, ditto, 3510 

Example. Required the horse power of an engine that will 
saw 368 planks, each being 30 feet by 2 feet 6 inches, in twelve 
hours. 

There are 29000 units of work done in sawing one square foot ; 

Then 30 x 2-5 x 368 x 29000 = units of work done in sawing 
the planks. 

Put X = tlie horse power of the engine ; 

Then 60 x 12 x 33000 x x = units of work done by the engine 
in twelve hours. 

^ 30 X 2-5 X 368 x 29000 ^„ . , 

Hence, x = = 33*7 horse power. 

^ 60 X 12 X 33000 ^ 



56 On WoRTi, 



Example. How many tons of coals would two men raise, work- 
ing with a wheel and axle, from a pit whose depth is 20 yards, in 
12 hours? 

From the Table, a man working with a wheel and axle will do 
2600 units of work in one minute. 

Then, 2600 x 60 x 12X2 = work done by the two men. 

Put X = the tons of coals raised. 

Then, 2240 x 20 x 3 x a; = work done by the two men. 

2600 X 60 X 12 X 2 ^^ „ , 

.\x = • =: 27 'So tons raised. 

2240 X 20 X 3 

The Traction of Horses at various rates of Travelling, 
It is a well known fact, that the traction or force which a horse 
can exert decreases with the increase of speed. 

Rate in miles per hour, 2 3 3|- 4 4|- 5 

Force exerted by the horse, 166 lbs. 125 104, 83, 62f, 41f. 

Accumulated Work. 

If a force be applied to move a body subject to no resistance 
whatever, it will be wholly occupied in increasing the speed of the 
body. In this case the work which is done by the action of the 
force applied is accumulated in the body, therefore it is called accu- 
mulated work. 

Put V = the velocity of the body or feet per second. 

And W = the weight of the body in pounds. 

Accumulated work = . 

64 

If W be measured in tons, and V be measured in miles per hour, 

Accumulated work = W V^ 

45 

A railway train 80 tons moves uniformly at the rate of 30 miles 
per hour, find the accumulated work. 

A 1^1 1 S388 X 80 X 900 ^,^^^^^ 

Accumulated work = = 5420800. 

45 

rru I. r ^x. ' 5420800 ,^^ , 

ihe horse power of the enffme = = 164 nearly. 

^ ^ 33000 -^ 

77 WV^ 

Generally the horse power of the engine = r ■ where W 

is in tons and Fin miles per hour. 33750 

The friction of a railway train is from 8 to 10 lbs. per ton. 

Work done by Machines. 
Tlie moving power, which is applied to any machine moving uni- 
formly, is employed in overcoming the resistance of friction, and 
useful work done at the working points of the machine. Hence, 



On Work. 57 

- . . _..^ 

the aggregate number of units of useful work yielded by any 
machine at its working point is less than the number received upon 
the machine directly from the moving power, by tlie number of 
units expended upon the resistance of friction. (The machine 
moving uniformly.) 

General Rule to find the Work done by any Machine. 

Find the distance through which the power (P) applied to the 
machine has travelled in one minute, and let this distance be 
called (a). 

Find the distance through which the weight ( W), producing 
useful work, has travelled in one minute, and let this distance 
be (6). 

Then a P — h W^= work done by friction per minute. 

And a P ^^ work applied per minute. 

b W= useful work done per minute. 

The Horse Power of an Engine. 

Let P be the mean effective pressure of the steam on the piston. 
I be the length of the stroke in feet. 
n be the number of strokes per minute. 

.*. Horse power of the engine == • •. 

^ ^ 33000 

^ oil 
The nominal horse power = -r^-x — as adopted by the Admiralty. 

On the Strength of Animals. 

Let P be the force in lbs. that any animal can exert when 
moving at {v) miles per hour. 

Put K = the greatest eftbrt the animal can exert when standing. 

And c = the greatest number of miles per hour the animal can 
give itself when unimpeded by any weight. 

V \ 

According to Bouguer, P = (I ;- I . K. 

Euler, P = {l---^\ . K 

Euler, P = (1 - -^^ . K 

It is readily seen that (v) miles per hou^ is equal to (88 v) feet per 
minute. Put L^the units of work done by the animal per minute, 

^'\ 
Then, according to Bouguer, U = 88 (v — — ^ j . X. 



58 The Crane. 



According to Euler, U = HS(v — —^\ . K. 
r, U=SSv{l-—^ 



Euler, 



The values of U will be the greatest wheu 

V — — . According to Bouguer. 



c 

€ 

v = -. 



Euler. 



Euler. 



Substitute these values in the formula for P and U, then there 
will result : 

— = the load of the animal when producing the greatest eflfect. 



2K 


(( «( it 


3 




4K 


€1 •< (i 


9 






22 c K=z the greatest effect, by first formula. 




— — =r = " . by second formula. 

^ V 3 




36 2 ck ^^ 

^rj — by third formula. 



T® Calculate tlie Different Parts of a Crane 
a§ respects Meclianical Advantage. 

(1.) The niimher of revolutions of the pinion to one of the wheel, 
the length of the handle, and the force applied being given, to 
find the diameter of the barrel. 

Rule. Multiply the diameter of the circle described by the 
; winch, or handle, in inches, by the power applied in lbs, and by 
j the number of revolutions of the pinion to one of the wheel ; divide 
I this product by the weight to be raised in lbs., and the quotient is 
: the diameter of the barrel in inches. 



EQUILIRRrUM AND PrfsRITvE OK BeaMS, 59 



(2.) The diameter of the barrel, the length of the handle, and the force 

applied given, to find the nwnber of revolutions of the pinion 

to one of the tvheel. 

Rule. Multiply the weight to be raised in lbs. by the diameter 

of the barrel in inches, and divide the product by the diameter of 

the circle described by the handle in inches, multiplied by the 

power applied in lbs., and the quotient is the revolutions of the 

pinion to one of the wheel. 

(3.) The diameter of the barrel, the number of revolutions of the pinion 

to one of the wheel, and the power applied given, to find the 

length of the handles. 

Rule. Multiply the weight to be raised in lbs. by the barrel's 

diameter in inches, and divide the product by the power applied in 

lbs., multiplied by the number of revolutions of the pinion to one of 

the wheel, and half the quotient is the length of the handles. 

(4.) The diameter of the barrel, the revolutions of the pinion to one of 

the wheel, and length of handles given, to find the power 

required. 

Rule. Multiply the weight to be raised in lbs. by the diameter 

of the barrel in inches, and divide the product by the diameter of 

the circle described by the handle multiplied by the revolutions 

of the pinion to one of the wheel, and the quotient is the power 

applied. 

The handles of a crane should not be less than ^ feet 11 inches or 3 
feet from the ground, and the jib to stand at an angle of about 45 
degrees. 

XTquilibrium and Pressure of Beams. 

The Parallelogram of Forces. 

It has been proved by experiment that three forces, proportional 
to the two sides of a parallelogram and its diagonal, are in a state 
of equilibrium when their directions are in the direction of these 
lines. 

Let two forces, represented in direc- 
tion and magnitude by the lines A 
and BO, act at the point 0, then a 
third force 00 in direction and magni- 
tude can be found, so that the three 
forces are in a state of equilibrium. 

Draw AD, BD, parallel to B, A, ^ 

respectively ; join T> 0, and produce it /^ 

to O, malting CO equal to D, then ^^ 
C is the force required. ""^^---., 

The two forces A 0, BO are called 
components, and 6^ the resultant of the 



(50^ 



Equilibrium and Pressure ok Beams. 



components. The components and resultant are called the paral-j 
lelogram of forces. | 

Any resultant force can be readily decomposed into two compo- i 
nents, which will be the sides of a parallelogram whose diagonal is 
the resultant. 

Let the beams E, B sustain a weight 
W) tons at the point ; draw Oi> verti- 
cal, and make it equal to {W) inches, then 
draw DA,DB parallel to 6> i^ and £ 
respectively ; measure I) A, D B in inches 
which will be the pressure in tons in the 
directions F and E. 




In this case JS'i^is a tie beam to 
prevent the lower ends of the 
beams E, OF from spreading. 
Draw D vertically equal to ( W) i 
inches, then draw DAyDB pa-! 
rallel to F, E and aA, b B,\ 
parallel to E F, then AD will he 
the thrust in F, and D B in E, and A a equal to bB will be i 
the thrust in the direction of the tie beam E F . i 



Draw OD vertically equal to (TF) 
inches, and draw i> ^~ parallel to OF, 
and D B parallel to E, then OB, OA 
will represent the pressures in the direc- 
tions OF, OE, 



Let ^ ^ be a beam whose centre of gravity 
is 0, and resting against an upright wall 
BE, the lower end resting on an abutment 
cut in the beam A E at A. 

Through the centre of gravity O draw 
the line C B vertically equal to the weight 
of the beam, draw B C, i> i^ parallel to EA, 
join CA' then OF represen-s the thrust at A in the direction 
CF, and FJD represents the thrust at B, and also the horizontal 
thrust at ^. 

To Compute the Tensioii of the ' guise ' and Shear-leg of a pair of 

Shears. 
Let BChe the shear-leg and A C the guise, and (IT) weight 
in tons supported at C, 




Specific Gravity. 61 



Make C as many inches as 
( W) contains tons, draw D E 
parallel to A G, then DE 
measured in inches will be 
the tension in tons of the 
guise A C, and C^ measured 
in inches will be the pressure 
in the direction of the shear- 
leg CE. 

To Compute the Tension on the guise arithmetically. 

V\xt AB-c, B C - a, and A C= h. 

, . b (6^ -a"- C-) W 
Then, tension in AC 




And the pressure in GB ~ 



V{a'\-h-\-c){h+c-a){a-{'C-h){a + h-c) 
a {IP+c'-a') W 



2c V{a-{-b'\-c){a-\-b—c)[b-\-c~a){a'\'C—6) 



SPECIFIC GEAYITY. 

The comparative density of various substances, expressed by the 
term Specific Gravity, affords the means of readily determining the 
bulk from the known weight, or the weight from the known bulk ; 
and this will be found more especially useful, in cases where the 
substance is too large to admit of being weighed, or too irregular in 
shape to allow of correct measurement. The standard with which 
all solids and liquids are thus compared, is that of distilled water, 
one cubic foot of which weighs 1000 ounces avoirdupois; and the 
specific gravity of a solid body is determined by the difference 
between its weight in the air and in water. Thus, 

If the body be heavier than water, it will displace a quantity of 
fluid equal to it in bulk, and will lose as much weight on immer- 
sion as that of an equal bulk of the fluid. Let it be weighed first, 
therefore, in the air, and then in water, and its weight in the air be 
divided b}* the difference between the two weights, and the 
quotient will be its specific gravity, that of water being unity. 

Example. A piece of copper ore weiglis 56i ounces in tlie air, 
and 43| ounces in water : required its specific gravity. 

56-25 — 43-75 — 12-5 and 56*25 -^- 12*5 = 4*5, the specific gravit}^ 

If the body be lighter than water it will float, and displace a 
quantity of fluid equal to it in 7vclght, the bulk of which will be 
equal to that only of the part iminersed. A heavier substance 



62 Specific Gravity* 



must therefore be attached to it, so that the two maj sink in the 
fluid. Then, the weight of the lighter substance in the air must be 
added to that of the heavier substance in water^ and the weight of 
both united, in water, be subtracted from the sum ; the w^ eight of 
the lighter body in the air must then be divided by the difference, 
and the quotient will be the specific gravity of the lighter substance 
required. 

Example. A piece of fir weighs 40 ounces in the air, and, being 
immersed in water attached to a piece of iron w^eighing 30 ounces, 
the tico together are found to weigh 3*3 ounces in water, and the 
iron alone 25'8 ounces in the w^ater : required the specific gravity 
of the wood. 

40 + 25-8 = 65.8 ~ 3-3 = 62-5 ; and 40 -r- 62*5 = 0'64, the spe- 
cific gravity of the fir. 

The specific gravity of a fluid may be determined by taking a 
solid body, heavy enough to sink in the fluid, and of known specific 
gravit}^, and weighing it both in the air and in the fluid. The dif- 
ference between the two weights must be multiplied by the specific 
gravity of the solid body, and the product divided by the weight 
of the solid in the air ; the quotient Avill be the specific gravity of 
the fluid, that of water being unity. 

Example. Required the specific gravity of a given mixture of 
muriatic acid and water ; a piece of glass, the specific gravit}' of j 
which is 3', weighing 3| ounces when immersed in it, and 6 ounces 
in the air. 

6 — a-'Fo = 2-25 X 3 = G'To -f- 6 = 1-125, the specific gravity. 

Since the weight of a cubic foot of distilled water, at the tempe- 
rature of 60 degrees (Fahrenheit), has been ascertained to be 1000 
avoirdupois ounces, it follows that the specific gravities of all bodies 
compared with it, may be made to express the weight, in ounces, 
of a cubic foot of each, by multiplying these specific gravities 
(compared with that of water as unity) by 1000. Thus, that of 
water being 1, and that of silver, as compared with it, being 10-474, 
the multiplication of each by 1000 will give 1000 ounces for the 
cubic foot of water, and 10474 ounces for the cubic foot of silver. 



Table of Specific Gravities — Water = 1000, 



Metals. 

Antimony, . . . 6-712 

Zinc, 7-100 

Cast Iron,. . . . 7*207 

i Tin, T'291 

Steel, 7-816 

Cast copper, . . . 8-7S8 



Mercury, .... 13-5S6 Crown glass, . . . 2*488 

Oraanic Bodies j Flint glass, . . . 3-329 

Ui ganiG Moaies. Eock crystal, . . 2-658 

Oak wood, . . . 925 Diamonds, . . 3-501 

Cork, 240 ' .^ 

Ivory, 18261 Liquids. 

White wax, . . . 960 i Ether, 0-715 

' Alcohol, .... «-79« 



Bismuth,.' . . 9 -882 I ^^^6^^^^^ ^'^"^-^•^^^^^^'^ j Oil of turpentine, . 0*870 

Silver, 10*474 i Bodies. ! Sea water, . . . 1*C26 

Lead, 11-852 i Agate, 2-590 i Milk, 1-030 

Gold, 19-258 i Amber, 1*078 ' Nitric acid, . . . 1-508 

Platinum, .... 20-837 1 Sulphur, .... 2*033 i Sulphuric acid. . . 1-S45 



The Mecoanical Powers. 63 



Weights of given hulks of water and air for calculating the absolvie 
weights from the specific gravities of bodies. 

Cubic inch of distilled water (bar. 30, therm. 62) 

in grains 252*458 

. foot .,.,.in ounces avoir. 997-1869691 

. . , ..in pounds do. 62-3210606 

Weight of 100 cubic in. of air in grains do. 30-49 



Logaiitlinis, 

2-40219 
2-99875 
1-79463 
1-48416 



THE MECHANICAL POWERS, AI^D THEIR 
APPLICATION, 

The simple Mechanical Powers are six in number, viz. the Lever, 
the Pulley^ the Wheel and Axle, the Inclined Plane^ the Wedge, and 
the Screw. All machines are formed by combinations to a greater 
or less extent of these six elements. The mechanical effects, how- 
ever, of the whole, are ultimately resolvable into that of the lever. 

By means of the Meclianical Powers a great weight may be sus- 
tained, or a great resistance slowly overcome, by the application of 
a small force. Or, a great velocity may be imparted to a small 
weiglit or resistance, by the use of a great force or power. 

Tlie I^evera 

Levers are of three orders : 

In the first order, the fulcrum is between the weight and the 
power, 

Jn the second order, the weight is between the fulcrum and the 
power 

In the third order, the power is between the weight and the 
fulcrum. 

The bent lever has no peculiarity except that of form, which is 
given to it for convenience in use. Its properties are those of the 
first order. 

In order to preserve an equilibrium between the power and the 
weight, they must be to each other inversel}' as their distances from 
the fulcrum. 

Case 1. When the I^ever is of the first order, or lohen the fulcrum 
is between the power and the weight. 

EuLE. Divide the weiglit to be raised by the power to be applied; 



64 The Mechanical Powers. 



the quotient will give the difference of leverage necessary to support 
the weight in equilibrio. Hence, a small addition either of lever- 
age or weight will cause the power to preponderate. ■ 

Example 1. A ball weighing 3 tons is to be raised by 4 men, who 
can exert a force of 12 cwt. Required the proportionate length of 
lever. ^^ 

8 tons =: 60 cwt. ; and — ■ = 5. 
12 

In this example, the proportionate lengths of the lever to main- 
tain the weight in equilibrio, are as 5 to 1. But, although the ball 
is sustained by a force of only one fifth of its weight, no power is 
gained, for the weight j^asses through onh^ one fifth of the space 
passed through by the power. 

Example 2. A weight of 1 ton is to be raised with a lever 8 feet 
in length, by a man who can exert, for a short time, a force of rather 
more than 4 cwt. Required at wdiat part of the lever the fulcrum 
must be placed. 

20 cwt. 



- rrr o ; I. e. 



the weight is to the power as 5 to 1 ; therefore, 



4 cwt. 

ft 

= 1 foot and a third from the weio^ht. 



5x1 



Example 3. A weight of 40 lbs. is placed one foot from the ful- 
crum of a lever. Required the power to raise the same when the 
length of the lever on the other side of the fulcrum is five feet. 

40 X 1 ^ ,^ ^, 

• — r=z 8 lbs., the power. 

o 

Case 2. When the lever is of the second order, or when the fulcrum 
is at one end of the lever and the 'power at the other, with the weight 
between them. 

Rule. As the distance between the power and the fulcrum is 
to the distance between the weight and the fulcrum, so is the effect 
to the power. 

Example 1. Required the power necessary to raise 120 lbs. when 
the weight is placed six feet from the power and two feet from the 
fulcrum. 

As 8 : 2 :: 120 : 30 lbs., the power. 

Example 2. A beam 20 feet in length, and supported at both 
ends, bears a weight of two tons at the distance of eight feet from 
one end. Required the weight on each support. 

^ ^^ =16 cwt. on the support that is furthest from the 

20 feet ^^ 

40 X 12 

weight ; and • — =: 24 cwt. on the support nearest to the weight. 

20 leet 



TiiJE Mechanical Powers. 65 



Case 3. When the lever is of the third order, or the weight is at o?ie 
end of the lever, the fulcrum at the other, and the •power is applied 
between them. 

Rule. As the distance between the power and the fulcrum is 
to the length of the lever, so is the weight to the power. 

Example. The length of the lever being eight feet, and the weight 
at its extremity 60 lbs., required the power to be applied six feet 
from- the fulcrum to raise it. 

As 6 : 8 :: 60 : 80 lbs., Ans. 



TSie Fiilley, 

Pulleys are of two kinds, fixed and movable. 

The fixed pulley affords no economy of power, but merely changes 
its direction. The movable pulley changes its position with that of 
the weight, and effects a saving equal to half the power. An equi- 
librium is preserved between the power and weight, when the 
w^eight is equal to the product of the power and twice the number 
of movable pulleys. 

Pule. Divide the weight to be raised by twice the number of 
pulleys in the lower block ; the quotient will give the power neces- 
sary to raise the Aveight. 

Example. Eequired the power to raise 600 lbs. when the lower 
block contains six jDulleys. 



600 
6 X ii 



- rrz 50 lbs., the power. 



The Wlaeel and Axle. 

The wheel and axle act as a revolving lever ; and in order to 
obtain an equilibrium between the power acting on the circum- 
ference of the wheel, and the weight or resistance acted on by the 
circumference of the axle, the power must be to the weight as the 
radius of the axle is to that of the wheel. One or more radii of the 
wheel, or winches, are often substituted for the wheel in the simple 
machine ; and in compound machines the action is communicated 
by teeth or cogs, forming wheel-and-pinion work. 

Rule. As the radius of the wheel is to the radius of the axle, so 
is the effect to the power. 

Example. A weight of 50 lbs. is exerted on the peripher}^ of a 
wheel wliose radius is 10 feet. Required the weight raised at the 
extremity of a cord Avound round the axle, the radius being 20 
inches. 

50 lbs. X 10 feet x 12 inches „ , . , , 
__ _ 300 lbs., the weiiirht 

20 inches. 

6* ' 



66 The Mechanical Powers. 



Tlie Inclined Plane. 

The inclined plane acts as a mechanical power by sustaining a 
portion of the weight to be raised, w^hile the direction of the 
applied force is changed from the perpendicular to one more or less 
horizontal, and the weight moves upwards on it in a diagonal 
between them. Equilibrium is sustained when the power is to the 
weight as the perpendicular height of the inclined plane is to its 
inclined length or hypothenuse, when the power acts in a direction 
parallel to the inclination of the plane ; but as the height is to the 
base when in a direction parallel to the base. 

Rule. As the length of the plane is to its height, so is the weight 
to the power. 

Example. Required the power necessary to raise 540 lbs. up an 
inclined plane 5 feet long and 2 feet high. 

As 5 : 2 :: 540 : 216 lbs., the power. 

The length, in the above rule, must represent that of the inclined 
surface, or of the base, accordingly as the power acts parallel to 
either of these surfaces. 



Tlie Wedge. 

The wedge may be regarded as two inclined planes, united by a 
common base, acting on two weights or resistances at once, or on a 
fulcrum and a weiglit, between which it moves, generall}', in prac- 
tice, by the impulse of successive blows. 

As in the inclined plane, equilibrium consists in the power being 
to the resistance as the back of the wedge is to its length, or to the 
length of its side, accordingly as the resistance acts perpendicularly 
to the central line of length or to that of the side. 

Case 1. When two bodies are forced from one another by means of 
a wedge, in a direction parallel to its back. 

Rule. As the length of the wedge is to half its back or head, so 
is the resistance to the power. 

Example. The breadth of the b^ck or head of the wedge being 8 
inches, and the length of either of its inclined sides 10 inches, 
required the power necessary to separate two substances with a 
lorce of 150 lbs. 

As 10 : U :: 150 : 22^ lbs., the power. 

Case 2. When only one of the bodies is movable. 

Rule. As the length of the wedge is to its back or head, so is the 
resistance to the power. 

Example. The iDreadth, length, and force, the same as in the last 
example. 

As 10 : 3 :: 150 : 45 lbs., the power. 



The Screw. 6*7 



Tlie Screw. 

The screw is an inclined plane, and may be supposed to be gene- 
rated by wrapping a triangle, or an inclined plane, round a cylinder. 
The base of the triangle is the circumference of the cylinder; its 
height, the distance between two consecutive cords or threads ; and 
the hypothenuse forms the spiral cord or inclined plane. 

Rule. To the square of the circumference of the screw, add the 
square of the distance between two threads, and extract the square 
root of the sum : this will give the length of the inclined plane. 
Its height is the distance between two consecutive cords or threads. 

When a winch or lever is applied to turn the screw, the power of 
the screw is as the circle described by the handle of the winch, or 
lever, to the internal or distance between the spirals. 

Case 1. When the weight to he raised is given, to find the poioer. 

EuLE. Multiply the weight by the distance between two threads 
of the screw, and divide the product by the circumference of the 
circle described by the lever. The quotient is the power. 

Example. Required the power to be applied to the end of a lever 
three feet long, to raise a weight of five tons with a screw of 1^ 
inch between the threads. 

11200 lbs. X 1-25 

-^ — ; — 61*9 lbs., the power. 

36 mches x 2 x 3-1416 ^ 

Case 2. When the power is given, to find the weight it will raise. 

Rule. Multiply the power by the circumference of the circle 
described by the lever, and divide the product by the distance 
betAveen two threads of the screw : the quotient will be the weight. 
The example is the converse of that in the former case. 



To Harden and Polish Alabaster. -.-1. Take a strong solution of 
alum, strain it, and put it into a wooden trough sufficiently large to 
contain the figure, which must be suspended in it by means of a 
thread of silk ; let it rest until a sufficient quantity of the salt is 
crystallized on the cast, then withdraw it, and polish it AA'ith a 
clean cloth and water. 

2. Take white wax, melt it in a convenient vessel, and dip the 
cast or figure into it ; withdraw, and repeat tlie operation of 
dipping until the liquid wax rests upon the surface of the cast; 
then let it cool and dry, when it must be polished with a clean 
brush. 



Toothed Wheels. 



TOOTHED WHEELS. 

The pitch (or the distance between the centres of two contiguous 
teeth) of cog-wheels is measured on the pitch-line^ or extreme cir- 
cumference of the wheel ; and the distance between that line and 
the centre of the circle is reckoned as the radius of the wheel. 

The following rules have been laid down for the diameters and 
number of teeth for wheels and pinions. 

KULE 1. 

As the number of teeth in the wheel + 2*25, 
Is to the diameter of the wheel, 
So is the number of teeth in the pinion + I'o, 
To the diameter of the pinion. 

Example. Given the number of teeth in the wheel = 210, the 
diameter of the wheel = 25 inches, and the number of teeth in the 
pinion = BO, to find the diameter of the pinion. 

As 210 + 2-25 : 25 :: 30 + I'S : 3-7102, — the diameter of the 
pinion. 

KULE 2. 

As the number of teeth in the wheel + 2 ''2 5, 

Is to the diameter of the wheel, 

So is (Ko. of teeth in pinion + No. of teeth in wheel) -v- 2, 

To the distance of their centres. 

Example. Given the number of teeth in the wheel = 210, the 
diameter of the wheel = 25 inches, and the number of teeth in the 
pinion = 30, to find the distance at which their centres should be 
placed. 

30 X '^10 ^ 

As 210 + 2-25 : 25 :: ~ — : 14-1342 inches, = the distance 

of their centres. 



On tlie Telocity of Wheels, I>ritiiis, Pulleys, 

Ac. 

When wheels are applied to communicate motion from one part 
of a machine to another, their teeth act alternately on each other ; 
consequentl}', if one wheel contains 60 teeth ar.d another 20, the 
one containing 20 teeth will make three revolutions, while the 
other makes but one ; and if drums or pulle3's are taken in place of 
wheels, the result will be the same, because their circumferences, 
describing equal spaces, render their revolutions unequal; from this 
the rule is derived, namely, 



Toothed Wheels. 69 



Multiply the velocity of the driver by the number of teeth it 
contains, and divide by the velocity of the driven : the quotient 
will be the number of teeth it ought to contain. Or, multiply the 
velocity of the driver by its diameter, and divide by the velocity of 
the driven: the quotient will be the diameter of the driven. 

If the velocities of driver and driven are given with the distance 
of their centres, 

Then the sum of the velocities : -! i -a. ^ j • !• *. : distance 

( velocity of driven J 

^ , ( radius of driven. 

01 centres : i j. ^ -, . 

( radius oi driver. 

Example 1. If a wheel that contains 75 teeth makes 16 revolu- 
tions per minute, required the number of teeth in another to work 
in it, and make 24 revolutions in the same time. 

^^75x10 , , 

Here = 50 teeth. = Ans. 

24 

Example 2. A wheel, 64 inches diameter, and making 42 revolu- 
tions per minute, is to give motion to a shaft at the rate of 77 revo- 
lutions in the same time ; required the diameter of a wheel suitable 
for that purpose. 

64 X 42 
Here — = 349 inches. = Ans. 

77 

Example 3. Required the number of revolutions per minute 
made by a wheel or pulley 20 in dies diameter, when driven by 
another of 4 feet diameter, and making 46 revolutions per minute. 

48 X 46 

Here ■ = 110-4 revolutions. = Ans. 

20 

Example 4. A shaft, at the rate of 22 revohitions per minute, is' 
to give motion, by a pair of wheels, to another shaft at the rate of 
15^; the distance of the shafts from centre to centre is 45 J inches; 
the diameters of the wheels at the pitch lines are required. 

22 X 45*5 

Here 22 + 15'5 : 22:: 45*5 iri. : = 26'69 in. 

22 + 15-5 

the radius of the driven wheel ; which, doubled, gives 53"38 inches, 
the diameter. 3= 1st Ans. 

Therefore 45*5 inches— 26'69 inches = 18 81 inches, the radius of 
the driver; which, doubled, gives 37*62 inches, the diameter. =2d 
Ans. 

Example 5. Suppose a drum to make 20 revolutions per minute, 
required the diameter of another to make 58 revolutions in the 
same time. 

Here 58 -=- 20 =r 2*9, that is, their diameters must be as 2-9 to 1 ; 
tlius, if the one making 20 revolutit)ns be called oU inches, the other 
will be 30 -v- 2 9 = 10'o45 inclies diameter. 



70 Toothed Wheels. 



Example 6. Required the diameter of a pulley, to make 12^ 
revolutions in tlie same time as one of 32 inches making 26. 

32x26 . , 

Here == 66'56 inches diameter. 

12*5 

Example '7. A shaft, at the rate of 16 revolutions per minute, is 
to give motion to a piece of machineiy, at the rate of 81 revolutions 
in the same time ; the motion is to be communicated bj^ means of 
two gearing wheels and two pulleys, w4th an intermediate shaft ; 
the driving wheel contains 54 teeth, and the driving pulley on the 
axis of the driven wheel is 25 inches diameter ; required the number 
of teeth in the other wheel, and the diameter of the other pulley. 

Let the driven wheel have a velocity of 36, a mean proportional 
between the extreme velocities 16 and 81 ; 

then, =: 24, the number of teeth in the driven wheel. = 

Ut Ans. 

36 X 25 

And ^^ = 11 -11 inches, diameter of the driven pulley. = 

81 ' r J 

2d Ans. 

Example 8. Suppose in the last example the revolutions of one 
of the wheels to be given, the number of teeth in both, and like- 
wise the diameter of each pulley, to find the revolutions of the last 
pulley. 

16 X 54 
Here - — — — == 36, velocity of the intermediate shaft. =:^ns. 

36 X '^5 
Also, — — ^^— rr: 81, the vclocity of the machine. 



Gold Lustre for Stone-ware. — Gold, 6 parts; aqua regia, 36 
, parts. Dissolve : then add, tin, 1 part. Next add balsam of sul- 
; phur, 3 parts ; oil of turpentine, 1 part. Mix gradually in a mortar, 
i and rub it in until the mixture becomes hard ; then add oil of tur- 
i pentine, 4 parts It is then ready to be applied to a ground 
• prepared for the purpose. 

I! To Petrify Wood, <fec. — Take equal quantities of gem-salt, rock- 
; alum, white vinegar, chalk, and pebbles powdered. Mix all these 
; ingredients: there will happen an ebullition. If, after it is over, 
i you throw into this liquor any porous matter, and leave it there 
'. soaking four or five days, they will positively turn into petrifactions. 



N 



Steam Power and the Steam- Engine. 71 



STEAM POWER AND THE STEAM-ENGINE. 

Steam is of great utility as a productive source of motive power ; 
in this respect, its properties are — elastic forco, expansive force, 
and reduction by condensation. Elastic signifies the whole urgency 
or power the steam is capable of exerting with undiminished effect. 
J^y expansive force is generally understood the amount of diminish- 
ing effect of the steam on the piston of a steam-engine, reckoning 
from that point of the stroke where the steam of uniform elastic 
force is cut off : but it is more properly the force Avhich steam is 
capable of exerting, when expanded to a known number of times 
its orio'inal bulk. And condensation, here understood, is the abstrac- 
tion or reduction of heat by another bod}^ and consequently not 
properly a contained property of the steam, but an effect produced 
by combined agency, in which steam is the principal ; because an^f 
colder body will extract the heat and produce condensation, but 
steam cannot b^ so beneficially replaced by any other fluid capable 
of maintaining equal results. 

The rales formed by experimenters, as corresponding with the 
results of their experiments on the elastic force of steam at given 
temperatures vary, but approximate so closely, that the following 
rule, because of being simple, may in practice be taken in prefer- 
ence t/o any other : 

Rule. To the temperature of the steam, in degrees of Fahrenheit, 
ad<l 100 ; divide the sum by 177 ; and the 6th power of the 
quotient will equal the force in inches of mercury. 

Example. Required the force of steam corresponding to a tem- 
perature of 312"". 

— = 2*B277^= 159 inches of mercury. 

1 

To Estimate the Amowit of Advantage Gained hy Usincf Steam i 
Expansively in a Steam- Engine. j 

When steam of a uniform elastic force is employed tliroughout | 
the whole ascent or descent of the piston, the amount of effect pro- ; 
duced is as the quantity of steam expended. But let the steam be j 
shut off at any portion of the stroke — say, for instance, at one half 
— it expands by degrees until the termination of the stroke, and , 
then exerts half its original force ; hence an accumulation of effect 
in proportion to the quantity of steam. 

RuLE.^ Divide the length 'of the stroke by the distance or space 
into which the dense steam is admitted, and find the hyperbolic 
logarithm of tlie quotient, to which add 1 ; and the sum isthe ratio 
of the gain. 

Example. Suppose an engine Avith a stroke of G feet, and the 



Steam Power axd the Steam-Engine. 



steam cut off when the piston has moved through 2 ; required the 

ratio of gain by uniform and expansive force 

6 _^ 2 = 3 ; iiyperbohc logarithm of 3 = 1*0986 + 1 = 2-0986, 
ratio of effect ; that is, supposing the whole effect of the steam 
to be 3, the effect by the steam being cut off at ^ = 2 '0986. 

Again, let the greatest elastic force of steam in the cylinder of an 
engine equal 48 lbs. per square inch, and let it be cut off from 
entering the cylinder when the piston has moved 4^ inches, the 
whole stroke being 18; required an equivalent force of the steam 
throughout the whole stroke. 

18 -V- 4-5 =: 4, and 48 -^ 4 = 12. 
Logarithm of 4 + 1 = 2-38629. 
Then 2*38629 x 12 = 28*635 lbs. per square inch. 

In regard to the other case of expansion, when the temperature 

is constant, the bulk is iuvei'sely as the pressure; thus, suppose 

j steam at 30 lbs. per square inch, required its bulk to that of original 

i bulk, when expanded so as to retain a pressure equal to that of the 

■ atmosphere, or 15 lbs. 

■ '- — = 3 times its orio-inal bulk. 

15 

It is because of the latent heat in steam, or water in an aeriform 
state, that it becomes of such essential service in heating, boiling, 
drying, &c. In the heating of buildings, its economy, efficiency, ■ 
and simplicity of application are alike acknowledged; the steam : 
being simply "^conducted through all the departments by pipes, by ! 
extent of circulation condenses— the latent heat being thus given to .. 
the pipes, and diffused by radiation. In boiling, its efficiency is | 
considerably increased, if advantage be taken of sufficiently inclos-j 
ing the fluid, and reducing the pressure on its surface, by means of j 
an air-pump. Thus, water in a vacuum boils at about a tempera- 1 
ture of 98° ; and in sugar refining, where such means are employed, ; 
the syrup is boiled at 150°. | 

The latent heat of steam at the common pressure of the atmo- \ 
sphere, according to very accurate experiments, is found to be | 
1000°; and we know that the sensible, or thermometric heat r= j 
212°. Now 212°- 32° = 180°, and 1000° + 180° ~ 1180° ; there- j 
fore, steam at 212° is simply highly rarified water, and contains j 
1180° of heat; hence, to find the latent heat of steam at any otjierj 
temperature, subtract the sensible heat from 1180°, and add 32° = 
the latent heat. 

Example. Required the latent heat of steam whose sensible heat 
is 224°. 

1180° — 224° = 956^ 
And 956° + 32° = 988° latent heat. 

A cubic inch of v/ater produces about 1700 cubic inches of steam 



Steam Powe-i and the Steam-Engine. 



at 212°, or the common pressure of the atmosphere; but the boiling 
point varies considerably with the pressure on the surface of the 
fluid ; thus, in a vacuum, water boils at about 90° ; under common 
pressure, at 212° ; and when pressed with a column of mercury 4 
inches in height, at 216° ; each inch of mercury producing by its 
pressure a rise of about 1° in the thermometer. 

Tiie pressure or force of steam in the boiler (less than the weight 
upon the safety-valve) is generally indicated by a column of mer- 
cury in a bent iron tube, which causes the range of the float to be 
only lialf the range of the mercury, 2 inches of mercury being 
nearly equal to 1 lb. pressure of steam in the boiler, thus : 



Each inch rise of the float indicates a pressure of nearly 1 lb. 



Level of the mercury when there is no force of steam above the 
pressure of the atmosphere. 



To Calculate the Effect of a Lever and Weight upon the 
Safety- Valve of a Steam-Boiler, d;c. 
The lever, under all circumstances, is balanced by a known weight 
or weights, on the short end, making its point of rest on the valve 
the centre of motion ; so that the weight, added to that of the 
lever, is the effective weight upon the valve, independent of any 
other additional weight, thus : 





i 



There are three different wa3^s that it may be required to calcu- 
late the lever : 

1. Wheti a certain pressure is required upon the valve, the distance 
of the loeight upon the lever, and the distance of the valve from the 
centre of motion given, to find \ohat weight loill be required upon 
the lever at that distance. 

From the pressure on the valve in lbs. subtract the weight of the 
valve in lbs. and the effective weight of the lever, multiply the 
remainder by the distance between the fulcrum and the valve, and 
divide the product by the distance between the fulcrum and tlie 



f!4: Steam Power and the Steam-Engine. 



weight, and the quotient is the weight in lbs. required to be placed 
upon the lever at that distance. 

2. When a certain pressure is required upon the valve, the weight 
upon the lever and distance of the valve from the centre of motion 
giveyi, to find where that weight must he placed. 

From the required weight upon the valve in lbs. take the weight 
of the valve, add the effective weight of the lever, multiply the 
remainder by the distance between the fulcrum and the valve, and 
divide tlie product by the weight in lbs. upon the lever, and the 
quotient is the distance in inches from the fulcrum that the weight 
must be placed. 

3. When the distance of weight, distance of valve from the centre 
of motion, and loeight upo7i the lever are given, to fnd what pressure 
is upon that valve. 

Multiply the weight in lbs. upon the lever by the distance in 
inches to the fulcrum, divide the product by the distance between 
the fulcrum and the valve, and the quotient, plus the weight of the 
valve and effective weight of the lever, equal the weight upon the 
valve in lbs. 

Example 1. Suppose the lever A B (as above) to be 24 inches in 
length, and the valve C placed 5 inches from the centre of motion 
A, what weight must be placed upon the lever 20 inches from A, to 
equal 80 lbs., on the valve C, the weight of the lever being 2 lbs., 
the weight I), which balances the lever, 4^ lbs., and the weight of 
the valve 3 lbs. ? 

2 lbs. weight of the lever. 
i.- 4'5 " to balance ditto. 

3 " weight of the valve. 

9-5 lbs. 



Then ?^ti^= 17-625 lbs. 

20 

Example 2. Suppose the weight upon the lever equal 17 '625 lbs., 
it is required at what distance from A the weight must be placed 
to equal 80 lbs. at C. 

80 -— 9-5 X 6 ^^ . , 

= 20 inches. 

17-625 

Example 3. Suppose, as before, that a weight of 17 '625 lbs. is 
placed upon the lever 20 inches from A, required the pressure at 
C, the distance from the centre of motion being 5 inches, and the 
effective weight of the lever at that point equal 6| lbs., also the 
weight of the valve 3 lbs. 

9''' 5 + 6-5 

fill.,..,.. + 3 

-"' ^"^ "•■ =: 80 lbs. 



Steam Power and the Steam-Engine. 



75 



To Find the Prosper Diameter for a Safety- Valve. 

Multiply the bottom surface of the boiler, or surface immediately 
exposed to the action of the fire, in feet, by the multiplier opposite 
to the pressure in lbs. on each square inch of the safety-valve, and 
the square root of the product is the valve's diameter in inches at 
the narrowest part If the boiler is to have two safety-valves, then 
the square root of half the product equal the diameter of each. 



Pressure in lbs. 

'per square inch. 

3 



4 
5 

6 • 

7 ■ 

8 . 
10 • 
12 ■ 



Multipliers. 

-356 

353 

-348 

-344 

339 

336 

329 

321 



Pressure in lbs. 
per square inch. 

15 

20 

25 

30 



40 

45 . 
50 . 



Multiplien 

315 

-305 

293 

-289 

'282 



•270 
•264 



In constructing steam-engines, the following simple rule for 
obtaining the nominal horse power is noAV generally adopted : 

The area of the cylinder in square inches multiplied by 7 lbs. 
pressure, multiplied into the speed of the piston in feet per minute, 
divided by 330uO, equal the nominal horse power. 



nominal H. P. 



Thus, area of cylinder x 7 lbs. X feet per minute _ 
33000 ~~~ — 

The length of stroke and relative speed of piston, and number of 
revolutions per minute, will be found by the following table. In 
calculating the gross horse power developed in any cylinder, as 
shown by the indicator, it has been customary to allow one-tenth, 
and sometimes one-eighth, for friction ; this is now very properly 
abandoned, and the following rule for calculating the indicator 
diagram should be always adopted : the mean pressure as shown 
on the card, multiplied into the area of the cylinder, multiplied 
into the speed of piston, in feet per minute, when the card was 
taken ; this product, divided by 33000, will give the gross or indi- 
cated horse power : 

ft. in. ft. 

For 3 stroke 30 revolutions per minute =180 per minute. 



3 6 




27 


" 


= 189 


4 




24i 


u 


= 196 


4 6 




22| 


" 


= 204 


5 




21 


i( 


= 210 


5 6 




19| 


(( 


= 216 


6 




18^ 


u 


= 222 


6 6 




171 


u 


= 226 


7 




16-^ 


(C 


= 231 


7 6 




15!J 


it 


= 236 


8 




15 


«< 


nr 240 


8 6 




u/v 


(( 


= 244 


9 




13| 




= 247 



^ Steam Power and the Steam-En gine. 

The Air-Pump. The diameter of the air-pump should be a little 
more than half the diameter of the cylinder, or the diameter of the 
cylinder in inches multiplied by -6 will give the diameter of the 
air-pump in inches, the length of stroke to be one-half the length 
of stroke of the piston. 

The Conde7iser should never be less than half the capacity of the 
cylinder ; and in engines where the pressure on the boiler ranges 
from twelve to twenty pounds on the square inch, a much larger 
condenser should be given. 

The foot and deliver}^- valve passages should have an area of one- 
third of the air-pump. 

The Steam-Ports. The area of the steam-ports on the cylinder 
should never be less than one-twentieth of the area of the cylinder. 
If the speed of the piston is above 250 feet per minute, the ports 
should never be less than one-fourteenth the area of the cylinders. 

The Cold- Water Pump. The capacity of the cold-water pump 
should be not less than one-thirty-sixth of the capacity of the 
cylinder. 

The Fly- Wheel. To find the weight of the fly-wheel rim the 
following practical rule is generally adopted: 

Horse power of the engine x 2000 
(velocity of circumference of wheel in feet per second) 2""" 
the weight of the fly-wheel in cwts. 

The Fly- Wheel, or Crank-Shaft. The nominal horse power of 
the engine and speed of the shaft being given, the diameter of this 
shaft, whether cast or wrought iron, will be found in the Tables of 
Strength of Shafts. 

The Governor. To find the number of revolutions, divide 375 bj- 
the square root of the length of the pendulum ; half of this quotient 
is the number of revolutions the balls ought to make per minute. 

To find the length of the pendulum, divide 375 by twice the 
number of revolutions ; the quotient squared is the length of the 
pendulum. 

Ge7i€ral Proportions of Locomotive Engines. 

For the area of the steam-ports when the stroke is 18 inches, the 
square of the diameter of the cylinder x -068 == the area in square 
inches. 

For the area of the eduction ports, the square of the diameter of 
the cylinder in inches x •128 = the area in square inches. 

The breadth of the bridges between the eduction ports and the 
induction = | inch and 1 inch. 

TJie diameter of the chimney = the diameter of the cylinder. 

For the area of the fire-grate, the diameter of the cylinder in 
inches x '77 == the area in superficial feet. 

For the effective heatiiig-surface of the boiler, the square of the 
diameter of the cylinder in inches x 5 -r- 2 = area in square feet. 



Steam Power and the Steam-ii^ngine. 



11' 



For the diameter of the feed-pump ram^ the square of the diameter 
of the cylinder in inches x 'Oil = the diameter in inches. 

For the cubical conte7it of the steam-romn, the square of the diame- 
ter of the cyUnder in inches x 9-r-40=content m cubic feet. 

For the cubical content of inside fire-box above fire-bars^ the square 
of the diameter of the cylinder in inches-T-4= content in cubic feet. 

For the inside diameter of the steam-pipe^ the square of the dia- 
meter of the cylinder in inches x '03 = the diameter in inches. 

For the diaineter of the branch steam-pipe, the square of tlie dia- 
meter of the cylinder in inches x '021 =: the diameter in inches. 

For the diameter of the top of the blast-pipe, the square of the dia- 
meter of the cylinder in inches x -017 = the diameter in inches. 

For the diameter of the feed-pipes, the diameter of the cylinder in 
inches x '141 := the diameter in inches. 

For the diameter of the piston-rod, the diameter of the cylinder in 
inches -^ 7 = the diameter in inches. 

For the thickness of the piston, the diameter of the cylinder in 
inches x 2 -r- 7 = the thickness in inches. 

For the diameter of the con7iecting-rod at the middle, the diameter 
of the cylinder in inches x '21 = the diameter in inches. 

For the diameter of the plain part and iriside bearing of the crank- 
axle, the cube root of the square of the diameter of the cylinder in 
inches x '96 = the diameter in inches. 

For the diameter of the outside bearings of the crank for axle, the 
cube root of the product of the square of the diameter of the 
cylinders in inches x -396 = the diameter in inches. 

For the diameter of the craiik-bearing, the diameter of the cylinder 
in inches x '404 = the diameter in inches. 

For the length of the crarik-bearing, the diameter of the cylinder 
in inches x '233 =. the length in inches. 

Jlemarks on Steam-Engine Boilers and their Proportions, 

For engines designed to give a gross indicator horse power of at 
least twice the nominal horse power, the grate surface should be 
•66 or '69 square feet per nominal horse power, but may be increased 
to '75 square feet, and should never be diminished to less than '60 
square feet as a minimum. 

The area of opening over the bridges or through the tubes, 
should be -125 square feet, or 18 square inches per horse power, 
and may be increased to '143 square feet, or 20 square inches with 
advantage, particularly in tubular boilers, and should never be 
diminished to less than 15 square inches, or -109 square feet per 
horse power. 

The area of chimney should be •076 square feet, or 11 square 
inches, but may be increased to 13 square inches, and should never 
be diminished to less than 10 square inclies per horse power. 



7* 



Steam Power and the Steam-Engine. 



The heating surface in fire-places and flues should be 14 square 
feet per horse power, exclusive of all bottom surface, but may be 
increased to 15 square feet, and should never be diminished to less 
than 12 square feet per horse power. 

In calculating tubular boilers the whole surface of the tubes 
should be taken, and there should be a total of 17 square feet per 
horse power in the fire-places and tubes. 

In engines designed to work to a gross power in the cylinder by 
the indicator greater than twice the nominal horse power, these 
proportions must be increased ; or, if the reverse be intended, they 
may be diminished in proportion. 



Of the Presmre of Steamy in Inches of Mercury, at Different 
Temperatures. 



I. 


IL 


III. 


lY. 


V. 


VI. 


Temperature, 












Fahrenheit. 


Dal ton. 


Ure. 


Young. 


Macneill. 


Tredgoid. 


0° 


0-08 










10 


0-12 










20 


0-17 




0-11 






32 


0-26 


0-20 


0-18 




0-17 


40 


0-34 


0-25 


0-20 




0-24 


50 


0-49 


0-36 


0-36 


0-36 


0-37 


60 


0-65 


0-52 


0-53 




0-55 


70 


0-87 


0-73 


0-75 


0-73 


0-78 


80 


1-16 


1-01 


1-05 




111 


90 


1-59 


1-36 


1-44 


1-36 


1-53 


100 


2-12 


1-86 


1-95 




2-08 


110 


2-79 


2-45 


2-62 


2-46 


2-79 


120 


3-63 


3-30 


3-46 




3-68 


130 


4-71 


4-37 


4-54 


4-41 


4-81 


140 


6-05 


5-78 


5-88 




6-21 


150 


7-73 


7-53 


7-55 


7-42 


7-94 


160 


9-79 


9-60 


9-62 




10-05 


170 


12-31 


12-05 


12-14 


12-05 


12-60 


180 


15-38 


1516 


15-23 




15-67 


190 


18-98 


19-00 


18-96 


18-93 


19 00 


200 


23-51 


23-60 


23-44 




23-71 


210 


28-82 


28-88 


28-81 


28-81 


28-86 


212 


30-00 


30-00 


30-00 


30-00 


30-00 


220 


35-18 


35-54 


35-19 




34-92 


230 


44-60 


43-10 


42-27 


42-63 


42-00 


240 


53-45 


51-70 


51-66 




60-24 



Steam Power and the Steam-Engine, 



79 



Of the Temperature of Steam at different Pressures in Atmospheres, 


I. 


11. 


III. 


lY. 


V. 


1 VI. 


Temperature 












Fahrenheit. 


French Acad. 


Dr. Ure. 


Young. 


Macneill. 


Tredgold. 


1st At. 


212-0° 


212° 


212° 


212° 


212° 


2d - 


250-5 


250-0 


240-3 


249 


250. + 


3d - 


275-2 


275-0 


271 




274.+ 


4th " 


293-7 


291-5 


288 


290 


294.+ 


5th '' 


308-8 


304-5 


302 




309. + 


6th " 


320-4 


315-5 






322.- 


7 th " 


331-7 


325-5 








8th *' 


342-0 


336-0 




337 


342.+ 


9th " 


350-0 


345- 








10th " 


358-9 








; 


11th " 


366-8 










12th " 


374-0 








372— 


13th " 


880-6 










14th " 


386-9 










loth '' 


392-8 










16th " 


398-5 










17th " 


403-8 










18th " 


408-9 










19 th " 


413-9 










20th " 


418-5 








414- 


30th " 


457-2 










40th " 


466-6 










60th " 


510-6 











To Prevent Spontaneous Combustion. — It is a fact better ascer- 
tained than accounted for, that fixed oils, wlxen mixed with any 
light kind of cliarcoal, or substances containing carbon, such as 
cotton, flax, or even wool, which is not of itself inflammable, heat 
by the process of decomposition, and after remaining in contact 
some time, at length burst into flame. This spontaneous combus- 
tion takes place in waste cotton which has been employed to wipe 
machines, and then thrown away and allowed to accumulate into a 
heap. We have known an instance of the kind in a manufactory 
for spinning worsteds, where the waste avooI, or "slubbings," as it is 
termed in Yorkshire, was thrown into a corner and neglected. It 
then heated, and was on the point of bursting into flame, when the 
attention of the workmen was directed to the heap by the smoke 
and smell. In cotton mills the danger exists in a still greater degree, 
and it is believed that the destruction of many cotton factories has 
been occasioned by this means. The cause of this peculiar property 
of fixed oils deserves more attention than has hitherto been paid to it. 



80 



Steam Powp:r and the Steam-Engine. 



TABLE 

Of the Elastic Force of Steam, and Correspo7iding Temperature of the 
Water with which it is in Contact. 



1 = 

a. 


o ■s ■ 
o = o 

1=1 


rt £ c 


Volume of 
Steam com- 
pared with 
the Volume 
of Water. 


il 

2^02 

0. 


Elastic Force 
in Inches of 
Mercury. 


ill 

III 


Volume of 
Steam com- 
pared with 
the Volume 
of Water. 


lbs. 

14-7 


30-00 


212-0 


1700 


Ib3. 

49 


99-96 


281-9 


564 


15 


30-60 


212-8 


1669 


50 


102-00 


283-2 


554 


16 


32-64 


216-3 


1573 


61 


104-04 


284-4 


544 


17 


34-68 


219-6 


1488 


62 


106-08 


285-7 


534 


18 


36-72 


222-7 


1411 


53 


108-12 


286-9 


525 


19 


38-76 


225-6 


1343 


54 


110-16 


.288-1 


616 


20 


40-80 


228-5 


1281 


55 


112-20 


2893 


508 


21 


42-84 


231-2 


1225 


56 


114-24 


290-5 


600 


22 


44-88 


233-8 


1174 


57 


116-28 


291-7 


492 


23 


46-92 


236-3 


1127 


58 


118-32 


292-9 


484 


24 


48-96 


238-7 


1084 


59 


120-36 


294 2 


477 


25 


51-00 


211-0 


1044 


60 


122-40 


295-6 


470 


26 


53-04 


248-3 


1007 


61 


124-44 


296-9 


463 


27 


55-08 


245-5 


973 


62 


126-48 


298-1 


466 


28 


57-12 


247-6 


941 


63 


128-52 


299-2 


449 


29 


59-16 


249-6 


911 


64 


130-56 


300-3 


443 


30 


61-21 


251-6 


883 


65 


132-60 


301-3 


437 


31 


63-24 


253-6 


857 


66 


134-64 


302-4 


431 


32 


65-28 


255-5 


833 


67 


136-68 


303-4 


425 


33 


67-32 


257-3 


810. 


68 


138-72 


304-4 


419 


34 


69-36 


259-1 


788 


69 


140-76 


305-4 


414 


35 


71-40 


260-9 


767 


70 


142-80 


306-4 


408 


36 


73-44 


262-6 


748 


71 


144-84 


307-4 


403 


37 


75-48 


264-3 


729 


72 


146-88 


308-4 


398 


38 


77-52 


265-9 


712 


73 


148-92 


309-3 


393 


39 


79-56 


267-5 


695 


74 


150-96 


310-3 


388 


40 


81-60 


269-1 


679 


75 


153-02 


311-2 


383 


41 


83-64 


270-6 


664 


76 


155-06 


312-2 


379 


42 


8568 


272-1 


649 


77 


157-10 


313-1 


374 


43 


87-72 


273-6 


635 


78 


159-14 


314-0 


370 


44 


89-76 


275-0 


622 


79 


161-18 


314-9 


366 


45 


91-80 


276-4 


610 


80 


163-22 


315-8 


362 


46 


93-84 


277-8 


698 


81 


165-26 


316-7 


358 


47 


95-88 


279 2 


586 


82 


167-30 


317-6 


354 


48 


97-92 


280-5 


575 


83 


169-34 


318-4 


350 




* 1 


"his include 


s the press 


ire of t 


he atmospht 


jre. 





Steam Power and tuv: Steam-Engin'e. 



81 



T A B L E— (Continued). 



Ok 


lib 


= S Si 

III 


Volume of 
Steam com- 
pared with 
the Volume 
of Water. 


a. 




Temperature 
in decrees of 
Fahrenheit. 


Volume of 
Steam com- 
pared with 
the Volume 
of Water. 


lbs. 
84 


171-38 


319-3 


346 


lbs. 
98 


199-92 


330-5 


301 


85 


173-42 


320-1 


342 


99 


201-96 


331-3 


298 


86 


175-46 


321-0 


339 


100 


204-01 


332-0 


295 


SI 


177-50 


321-8 


335 


110 


224-40 


339-2 


271 


88 


179-54 


322-6 


332 


120 


244-82 


345-8 


251 


89 


181-58 


323-5 


328 


130 


265-23 


352-1 


233 


90 


183-62 


324-3 


325 


140 


285-61 


357-9 


218 


91 


185-66 


325-1 


322 


150 


306-03 


363-4 


205 


92 


187-70 


325-9 


319 


160 


326-42 


368-7 


193 


98 


189-74 


326-7 


316 


170 


346-80 


373-6 


183 


94 


191-78 


327-5 


313 


180 


367-25 


378-4 


174 


95 


193-82 


328-2 


310 


190 


387-61 


382-9 


166 


96 


195-86 


329-0 


307 


200 


408-04 


387-3 


158 


97 


197-90 


329-8 


304 
















TAB 


LE 








Q 


f the Fore 


e and Tei 


nperatur 


e of Steam, in Atmospheres. 



Temp. Fah. 



De^. 

212-00 
250 52 
275-18 
293-72 
307-50 
320-36 
831-70 
341-78 
350-78 



10 
11 
12 
13 
14 
15 
16 
17 
18 



Temp. Fah. 



Deg. 

358-88 
366-85 
374-00 
380-66 
386-94 
392-86 
398-48 
403-82 
408-92 



19 
20 
21 
22 
23 
24 
25 



50 



Temp. Fah. 



Dcg. 

413-78 
418-46 
422-96 
427-28 
431-42 
435-56 
439-34 



510-60 



To WRITE ON Silver with a Black which will never go off. — 
Take burnt lead and pulverize it. Incorporate it next with sul- 
phur and vinegar, to tlie consistency of a painting color, and write 
with it on any silver plate. Let it dry, then present it to the fire 
so as to heat the work a little, and it is finished. 



Steam Power and the Steam-Engine. 



TABLE 

Of the Heating Power of various Comhustihle Substances, exhibiting 
the utntost Quantity of Water evaporated by the Given Weights, 
and the smallest Quantity of Air capable of producing Total Com- 
bustion. 



Species of Combustible. 



Pounds of i j 

Water j \ 

which a Pounds of I Weif ht of 

Pound can Boiling Wafer 'Atmospheric air 
heat, from evaporated by 1 at 32'\ to burn 1 
0" to 212'^. j Pound. Pound. 



Wood, in its ordinaiy state, 
Wood charcoal, .... 

Pit coal, 

Coke, 

Turf, 

Turf charcoal, .... 
Carburetted h3'drogeu, 

Oil, 

Wax, 

Tallow, 

Alcohol of commerce, . , 



26 

60 
65 
30 
64 
'76 

78 

52 



4-'72 
13-37 
10-90 
11 -81 

5-45 
11-63 
13-81 

14-18 

9-56 



4-47 
11-46 

9-26 
11-46 

4-60 

9-86 
14-58 

15-00 

11-60 



To Estimate Distance. — Observe how many seconds elapse 
between a flash of lightning and the thunder, and multiply them 
by 1142, the number of feet sound travels in a second, the product 
will be the distance in feet. The same process may be applied to 
the flash and report of a gun, or any other sound, provided we can 
ascertain the time at v/hich it is produced, and the interval that 
elapses before it reaches the ear. 

Illustration. Saw a flash of lightning five seconds before I heard 
the thunder: required the distance. 



5 X 1142 
3 X 1760~ 



^%^s mile distant. 



In the absence of a watch, the pulsations at the wrist may be 
counted as seconds, by deducting one from every seven or eight. 

Prismatic Diamond Crystals for Windows. — A hot solution of 
sulphate of magnesia, and a clear solution of gum arable, mixed 
together. Lay it on hot. For a margin or for figures, wipe off 
the part you wish to remain clear with a wet towel. 

Perfectly Black Hard Glass. — Plain paste, 600 parts ; zaffre, S 
parts ; manganese, 3 parts ; iron, 3 parts. 



Steam Powkr and the Steam-Engine. 













TABLE 














0-f Nominal Horse Power of Low 


Pressure Engines. 




Q c 
o c 








LENGTH OF STROKE IN FEET. 








1 


IK 


2 


2X 


3 


3K 


4 


4K 


5 


5K 


6 


7 


4 


•34 


•39 


•43 


•46 


•49 


•52 


•54 


■56 


•58 


•60 


•62 


-65 


5 


•53 


•61 


•67 


•72 


•76 


•8J 


•84 


'8S 


•91 


-94 


•96 


1-02 


6 


•76 


•87 


•96 


104 


no 


1*16 


1-22 


126 


1-31 


1-35 


1-39 


1 17 


7 


104 


119 


1-31 


1-41 


150 


1-58 


1-65 


1-72 


1-78 


1-84 


1-89 


199 


8 


1-36 


1-56 


172 


1-85 


1-96 


207 


2-16 


225 


2-33 


2-40 


2-47 


260 


9 


1-7^ 


197 


2-17 


234 


2-49 


2-62 


2-74 


2-84 


295 


3 04 


313 


330 


10 


2-13 


2-44 


2-68 


289 


308 


3 23 


3 38 


3-51 


3-64 


3-76 


387 


4 07, 


11 


2-57 


2-95 


3-24 


3-49 


3-77 


3-91 


415 


4 25 


4-40 


4 54 


4-68 


4-92 


12 


3 06 


351 


3-86 


416 


4-42 


465 


4-86 


506 


5-24 


5-41 


5-57 


5 86 


13 


3-60 


412 


453 


4-88 


5 19 


5-46 


5 64 


5'94 


615 


6-35 


6-53 


6 88 


14 


417 


4 77 


5-25 


5-66 


601 


6 33 


6 62 


6-88 


7-13 


7-36 


7-58 


7 98 


IS 


4-77 


5-48 


603 


650 


6 -90 


7-27 


760 


7-90 


819 


8-45 


8 70 


9 16 


16 


5-45 


6 23 


6 86 


7^39 


7-86 


8 27 


8-65 


8-99 


931 


961 


9-90 


10-42 


17 


615 


7*04 


775 


8 35 


8 86 


9-34 


9-76 


1015 


10-52 


10-85 


1117 


1176 


18 


6-89 


789 


8 68 


9-36 


9-94 


1047 


1094 


l'.-33 


1179 


12-17 


12'53 


13-19 


Id 


7-68 


879 


S-68 


10-42 


1117 


11-66 


1219 


1268 


1313 


13 56 


13 96 


14-69 


20 


8-51 


9-74 


1072 


1155 


12 27 


1292 


13-51 


14 05 


14-55 


1502 


15*46 


16-28 


22 


10-30 


11-79 


12-97 


13-98 


14-85 


1563 


16-62 


1730 


17 65 


18-18 


18-71 


19-70 


24 


12^26 


14 03 


15-44 


16 63 


17-67 


1861 


1945 


20-23 


20 95 


21-63 


22-27 


23-44 


26 


14-39 


16-46 


1812 


19-52 


20-75 


2184 


22 56 


23 75 


24 68 


25 39 


2614 


27'51 


28 


16-68 


1909 


2102 


22-64 


24 06 


25-33 


26-48 


2754 


28-52 


29-44 


3031 


3190 


30 


19-15 


2192 


24 13 


25 99 


2762 


29 07 


3040 


31-61 


3274 


33 80 


34-80 


36 63 


32 


2179 


24 96 


2751 


2957 


31 42 


3308 


34-59 


35-97 


37-26 


38-46 


39-59 


41-68 


34 


24 60 


23 16 


3099 


3339 


35-44 


37-34 


3904 


4060 


4206 


43-41 


44-69 


47-05 


36 


2757 


31-56 


34-74 


37 42 


39 77 


41-87 


43-77 


45 52 


47-15 


48-67 


50-11 


52-75 


38 


3072 


35- 17 


33-71 


4169 


44-66 


46 64 


48-77 


5072 


5254 


54-23 


55-83 


58-78 


40 


34 04 


33 97 


42 89 


46-20 


4910 


5169 


54 04 


56*20 


58-21 


6009 


61-86 


65*12 


42 


37 53 


42 96 


4729 


5094 


5413 


56 98 


5958 


61 96 


64-18 


66-25 


63-21 


71-78 


44 


4ri9 


47 15 


5190 


55-91 


59-38 


6254 


6646 


6800 


7044 


7271 


74-85 


78-79 


46 


45 02 


51-54 


5672 


61-10 


6488 


6819 


71-43 


74-33 


76-69 


T9'47 


8181 


8612 


48 


49 02 


56-11 


61-76 


66 54 


70-70 


74-42 


77-82 


80-94 


83-83 


86 53 


89'08 


93-78 


50 


53 19 


60-89 


67-02 


7219 


76 71 


80-76 


84-44 


87 82 


90 96 


93 89 


96-65 


101-7 


52 


57^55 


6586 


72-48 


78-08 


83 00 


87-35 


90 25 


94-98 


98-40 


101-55 


104-5 


no-o 


54 


62-04 


7102 


78-17 


84-20 


89-48 


94-20 


9849 


102-4 


106-1 


1095 


1127 


1187 


56 


66 72 


76 38 


84 07 


9055 


96-23 


10130 


1059 


1101 


1141 


117-8 


121-2 


1276 


58 


71-58 


81-93 


9018 


97 14 


103-2 


108 6 


1136 


118-2 


122-4 


1263 


129-2 


136-7 


60 


76 60 


87-68 


96 50 


1039 


110-4 


116-3 


121-6 


126-4 


1310 


135 2 


139 2 


1465 


62 


8179 


93-62 


103 04 


1110 


117-96 


124 18 


129-81 


13503 


13986 


144 37 


148-6 


156-7 


64 


87-15 


9984 


1100 


1183 


125-7 


132-3 


138 3 


1439 


149-0 


153-82 


158-4 


166 7 


66 


9268 


106-1 


1168 


125 8 


1336 


MO-7 


147-3 


153 


158-5 


163-6 


I68-4 


177-3 


68 


98 40 


112-6 


123-9 


133^6 


141-8 


1494 


1562 


1624 


168-2 


173 6 


178-8 


188-2 


70 


104-26 


119-3 


131-3 


141-5 


1504 


158 3 


1655 


172 1 


1782 


184 


189-4 


199-4 


72 


110 30 


1262 


1390 


1497 


159-1 


167-4 


175 1 


1821 


188-6 


194-7 


2OU-4 


2110 


74 


116-5 


1334 


1468 


158 1 


16r 9 


176 7 


185-4 


1924 


199-2 


205-7 


211-6 


223-4 


76 


122-9 


140-7 


1548 


166 8 


178-6 


1866 


1950 


2029 


2i0-l 


216-9 


223-3 


235 1 


78 


129-4 


1482 


1631 


1756 


186-7 


1965 


205-4 


2121 


221 -4 


2'2S-5 


235-2 


£47 6 


80 


136-2 


1558 


171 6 


184-8 


196-4 


2067 


2161 


224-8 


232-8 


240-4 


2474 


260 5 


82 


1430 


163-8 


180 2 


194 2 


206 2 


217-3 


226 9 


2378 


244 6 


^52 5 


260 


273 8 


84 


1501 


171-8 


1891 


2038 


2165 


227 9 


238 -3 


247-8 


256 7 


265 


272 8 


287 1 


86 


157-4 


180-1 


198 2 


2136 


227 


237-8 


24 7 4 


258-2 


2691 


2778 


286 


3010 


88 


164-8 


188 6 


207 6 


223 6 


237-5 


250 2 


26 16 


272 


281-7 


290-8 


299-4 


3152 


90 


1723 


1973 


2171 


233 9 


2486 


2617 


273-6 


284-5 


291-7 


304-2 


313-2 


329 7 


i 



























84 






Steam PowePw and tue Steam-Engine. 
















TABLE. 














Of Nominal Horse Power of High 


Pressure Engines. 




si 

C 








LENGTH 


OF STROKE IN FEET. 






i 

.! 

1 


1 


IX 


2 


2K 


3 


3X 


4 


4K 


5 


5K 


6 


7 


2 


•25 


•29 


•32 


■35 


•37 


•38 


"40 


•42 


•44 


•45 


•46 


-49 i 


2}i 


•39 


•45 


•50 


-54 


•57 


•60 


•63 


•66 


-68 


•70 


•72 


-76 1 


3 


•57 


•65 


72 


•78 


•83 


-87 


•91 


•95 


•98 


1 01 


ro4 


noi 


! 3>^ 


•78 


•89 


•98 


106 


ri3 


119 


1-24 


1 29 


134 


1-38 


1 42 


149, 


4 


102 


117 


129 


138 


147 


1-56 


1-62 


1-63 


1 74 


1 80 


1 86 


1-95! 


iVi 


1-29 


1-48 


163 


1-75 


1 86 


1-96 


205 


213 


221 


2-28 


235 


2-47; 


5 


1-59 


1-83 


201 


216 


2-28 


2-43 


252 


264 


2^73 


282 


2 88 


3-06 1 


5>'2 


1-93 


2 21 


2 43 


2 62 


2 78 


2 93 


312 


318 


3'30 


342 


3^51 


3 69i 


6 


2 28 


2-61 


2-88 


3 12 


3-30 


3 48 


3-66 


3-78 


393 


4^05 


417 


4-41 


6K 


2'69 


309 


339 


366 


3-90 


408 


4 23 


4-44 


4 62 


4-77 


4-89 


516 


7 


312 


357 


393 


4 23 


450 


4-74 


4 95 


5 16 


5 34 


5-52 


5 67 


5 97 


7K 


3-60 


4-11 


4 53 


4 86 


519 


5-46 


570 


5-94 


6 15 


6 S3 


651 


6-87 


8 


4 08 


4 68 


5-16 


5 55 


5 88 


6 21 


6-48 


6 75 


699 


7-20 


741 


7-80 


8K 


4 62 


5 28 


582 


6 '27 


6 63 


699 


7-32 


7 62 


7-89 


813 


8-37 


8-82 


9 


516 


5-91 


6 51 


7-02 


7-47 


7 86 


822 


8-52 


8-85 


9 12 


9 39 


990 


9K 


576 


6 60 


7 26 


7 80 


837 


8-76 


915 


951 


984 


10-17 


10-47 


10 01 


10 


639 


732 


804 


8 67 


921 


969 


10 14 


1053 


1092 


1128 


11 61 


12-21 


lOK 


705 


8-04 


8 88 


954 


1014 


10 68 


1116 


11 61 


12 03 


124S 


1278 


1347 


11 


7-71 


8-85 


972 


10-47 


11-31 


11 -73 


1245 


12-75 


1320 


136:. 


14 04 


14-76 


llX 


843 


966 


1062 


1146 


12 15 


12-78 


13 80 


13-92 


1461 


1491 


1533 


16 14 


12 


918 


1053 


11 58 


1241 


13-26 


13 95 


14 58 


1518 


15 72 


16 -22 


16-71 


1758 


V2^^ 


9-96 


1140 


12-57 


13^53 


14-37 


1515 


15 84 


1647 


1704 


17-58 


1812 


I9-O8 


13 


1080 


1236 


1359 


1464 


1557 


1638 


1692 


1782 


1845 


190E 


1959 


21 64 


isK 


1164 


1332 


1464 


15-78 


16-77 


17-67 


18-48 


1920 


19-89 


20-55 


21-15 


22 26 


14 


1251 


14-ol 


15 75 


16 98 


18 03 


1899 


1986 


20 64 


2139 


2208 


22-74 


23^94 


H'A 


1341 


1536 


16 92 


18-21 


19-35 


20-37 


21-30 


2214 


2295 


23 7C 


24-39 


25 62 


\ 15 


1431 


1644 


18-09 


1950 


20-70 


21 81 


2280 


23 70 


24-57 


25-3C 


26- 10 


27-48 


1 16 


16-35 


1869 


20-58 


22-17 


23-53 


24-81 


2595 


26-97 


27-93 


28 8c 


29-70 


3126 


i 17 


1845 


2ri2 


23 25 


22 05 


26 58 


28 02 


29-28 


30-45 


31 56 


32 55 


33 57 


35-28 


i 18 


20 67 


2367 


26 04 


28-08 


2982 


.31-41 


32 82 


34 14 


35 37 


3651 


37-59 


39-57 


i 19 


23 04 


26 37 


29 04 


3126 


3351 


34 98 


36-57 


38 04 


39 39 


40-68 


4188 


4407 


i 20 


25-53 


29-22 


32)6 


34-65 


3681 


38-76 


40 53 


42 15 


4365 


4506 


46 38 


48-84 


! 22 


30-90 


35-37 


38-9! 


4194 


44 55 


4689 


4986 


5190 


52-95 


54 54 


56 13 


59 10 


j 24 


36-78 


4-2-09 


46 -32 


49-89 


5301 


5583 


•58-35 


60 69 


6285 


64 6£ 


6681 


70 32 


26 


43-17 


4938 


54 36 


58-55 


6225 


65 52 


6768 


7125 


73 8 


761- 


78 42 


82 53 


28 


50 04 


57 27 


63 06 


6792 


72-18 


75 99 


7944 


82 62 


85 56 


88 -SS 


90 93 


95 70 


30 


57-45 


6576 


72-39 


7797 


82-86 


87-21 


9120 


94 83 


98-22 


101-4C 


104-4 


109-9 


32 


65-37 


74-88 


82 53 


88-71 


94 26 


99 24 


1037 


1079 


lire 


1154 


118-7 


1250 


34 


73-80 


8448 


92 9 


100 22 


106 3 


1120 


117 1 


1218 


126 2 


1302 


1340 


1411 


36 


82-71 


94 •68 


104 2 


1122 


1193 


125 6 


1133 


136 5 


141-4 


146 


1503 


158'2 


38 


92 16 


1055 


116 1 


125 


1340 


1369 


1463 


1521 


1576 


162 7 


1675 


1763 


40 


1021 


1169 


129-C 


1286 


1473 


155 1 


162- 1 


1686 


1746 


180 -2 


1856 


1S53 


42 


1126 


128 9 


1418 


152 8 


1624 


1709 


1787 


185 9 


1925 


1987 


204-6 


2153 


44 


1235 


1414 


155 7 


1677 


178 1 


1876 


199^4 


204 


211-3 


218 1 


224-5 


236*3 


46 


1350 


1546 


1701 


1833 


194 6 


204 -6 


214 3 


223 


230 


2.38 -4 


245-4 


258 3 


48 


1470 


168 3 


185-3 


199-6 


212 1 


223 2 


233-4 


242 8 


251 5 


259-6 


2672 


281-3 


50 


159 6 


1826 


2oro 


2165 


230- 1 


2423 


2533 


263-4 


27-. -9 


281 -6 


2899 


305 1 


52 


172 6 


197-6 


2174 


224-2 


249 


262 


270 7 


284 9 


2952 


304-6 


313 5 


330 


54 


186 1 


2130 


234 5 


252 6 


263 4 


282 6 


295 4 


307 2 


318-3 


3-28-5 


338 1 


356 1 


56 


200-1 


2991 


252 2 


271-6 


2387 


3039 


317 7 


330-3 


342-3 


353-4 


3636 


382-8 


i 58 


214 7 


245-8 


270 5 


2914 


309-6 


325 8 


340 8 


354 6 


3o7-2 


3789 


389-7 


410 1 


1 60 

1 


229 8 


263 


289 5 


3117 


3312 


3489 


364 -8 


379 2 


393 


40-5-6 


417-6 


4395 



Steam Powgu and the Steam-Engine. 



85 





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8«? 


Steam Power axd the 


Steam-Engine. 




TABLE 


Of the Revolutions per Mile of Driving Wheels, and Consvmption of 
Steam, and Water for each sized Wheel ; taking the steam admitted 
to each cylinder as exactly one cube foot, at a gross pressure of 
9Slbs. or Solbs. on the spring balance. 


WHEELS. 


Cylinder of Steam 

per Mile, and 

Consumption, 

taking Cylinder at 

one Cube Foot. 


Water per Mile, 

taking Steam at 

94lbs above 

atmosphere. 


Diameter. 


Circumference. 


Revolutions 
per Mile. 


feet. 
10 

9* 


ft. in. 
31 5 

29 101 


No. 

168 

176-9 


cube feet. 
672 


gallons. 
14-0 


707-6 


14-74 


9 


28 3i 


186-7 


746-8 


15 55 


H 


26 8| 


197-4 


789-6 


16-44 


8 


25 li 


210-1 


840-4 


17-5 


n 


23 6f 


224 


897-6 


18-69 


7 


21 111 


240 


960 


20-0 


6i 


20 5 


258-6 


1034 


21-5 


6 


18 9 


18 


280-5 


1122 


23-87 


H 


17 3 


33 


305-6 


1222-4 


25-45 


5 


15 8 


48 


336-3 


1344-4 


28-0 


^ 


13 11 


1 


379-0 


1493-6 


81-11 


4 


12 6 


92 


420-3 


1680-4 


35-0 


8| 


11 9 


37 


441-1 


1792-2 


37-33 


H 


10 11 


94- 


480-1 


1920-8 


40-0 


3 


9 5 


08 


560-2 


2240 


46-67 


Note- As there are two cylinders at work in a locomotive, consequently there are four 
cylinders of steam for each revolution. 


Modelling "Wax. — This is made of white wax, which is melted 
and mixed with lard to make it malleable. In working it, the 
tools and the board or stone are moistened with water, to prevent 
its adhering; it may be colored to any desirable tint with dry 
color. 



n 



Steam Power and the Steam-Engine. 



87 



TABLE 

Of Pressure of Steam, exclusive of that of the Atmosphere. 



PRESSURE. 


Tempera- 


PRESSURE, 
















Tempera- 


lbs. on 


Iti inches 




ture m de- 


lbs. on 


In inches 




ture in 


the 


of 


In atmo- 


grees of 


the 


of 


In atmo- 


degrees of 


sq. inch 


mercury. 


spheres. 


Fahrenheit 


sq. inch. 


mercury. 


spheres. 


Fahienhei 


1 


2*04 


•068 


213° 


51 


1C4^04 


3-468 


301O 


2 


4-08 


•136 


216 


52 


10608 


3 536 


302K 


3 


612 


•204 


219K 


53 


108-12 


3-604 


303X 


4 


816 


•272 


223 


54 


11016 


3-672 


l'U4>^ 


5 


10-20 


•340 


225X 


55 


11220 


3 740 


305^2 


6 


12-24 


•408 


228^2 


55 


114-24 


3 803 


30S>< 


7 


14 ^S 


•476 


231 


57 


116-28 


3-876 


307 X 


8 


16-32 


•544 


234 


53 


11832 


3944 


3083^ 


9 


18 36 


•612 


236 


59 


1-20 36 


4 012 


309 


10 


20-iO 


680 


239 


60 


122-40 


4080 


310 ) 


11 


22 44 
2448 


•748 
•816 


241 
243 


61 


124-44 


4-148 


311 


12 


62 


126 48 


4216 


312 


13 


26 52 


-884 


245^ 


63 


128-52 


4-2S4 


313 


14 


23-56 


•952 ^ 


247K 


64 


13056 


4 352 


314 


15 


30-60 


1020 


249K 


65 


13260 


4-4'^0 


315 


16 


32-64 


1 038 


251 K 


66 


134-64 


4-488 


316 


17 


34-63 


1 156 


253)^ 


67 


13663 


4 556 


317 


18 


36 72 


r224 


2553^ 


68 


138 72 


4 624 


317>^ 


19 


33-76 


1292 


257 


69 


140 76 


4 692 


318>^ 


20 


4080 


1 360 


259 


70 


14280 


4 760 


319 


21 


4 J -84 


1'423 


261 


71 


144 84 


4-828 


320 


22 


i\'Si 


1-496 


262K 


72 


146 88 


4-895 


321 


23 


46 92 


1 5S4 


264 


73 


148-92 


4 964 


322 


24 


48 95 


1632 


266 


74 


15096 


5 032 


322^ 


25 


5100 


1 700 


2673^ 


75 


153 00 


5100 


323K 


26 


53-04 


1-768 


269 


76 


155 04 


5-163 


324 


27 


55 08 


r836 


270K 


77 


157 08 


5 -20 6 


325 


28 


57 12 


1904 


272 


78 


15912 


5 304 


326 


29 


59-16 


r972 


273X 


79 


16116 


5372 


327 


30 


61-20 


2040 


275 


80 


163-20 


5-440 


3273^ 


31 


63 24 


2- 108 


2763^ 


81 


1 65^24 


5 508 


328 


32 


65 28 


2- 176 


278 


82 


16728 


5 576 


329 


33 


67 32 


2 244 


279 


83 


16932 


5-644 


330 


34 


69 36 


2-312 


2803^ 


84 


17136 


5 712 


330}4 


35 


71-40 


2-380 


282 


85 


173-40 


5 780 


331 


36 


7344 


2448 


283 


86 


175 44 


5-848 


332 


37 


75-43 


2516 


284 K 


87 


177-48 


5-916 


333 


38 


77 52 


2*584 


286 


83 


17952 


5 984 


3333^ 


39 


79 56 


2652 


287 


89 


18156 


6 052 


334 


40 


8160 


2720 


288 


90 


183 60 


6 120 


335 


41 


8:r64 


2 788 


289 


91 


185 64 


6-188 


335X 


42 


S568 


2-856 


290K 


92 


18768 


6 256 


336 


43 


87 72 


2921 


292 


93 


189 72 


6-324 


337 


44 


89-76 


2*992 


293 


94 


19176 


6 392 


338 


45 


91 80 


3-060 


294 


95 


193 SO 


6460 


338X 


46 


9 J -84 


3- 123 


295M 


96 


195-84 


6 528 


339 


47 


95-88 


3- 196 


297 


97 


197-88 


6-596 


340 


48 


97-92 


3264 


29S 


98 


199-92 


6664 


.S40K 


49 


99 96 


3-332 


299 


99 


20196 


6732 


341 


50 


10200 


3400 


300 


100 


204 00 


6800 


342 



LiNSKKD Oil, Clarified, for Varnishes. — Heat in a copper boiler 
50 gallons oflinseed oil to 280° Fall.; add 2^ lbs. of calcined white 
vitriol, and keep the oil at the above teniperatiire for half an hour; 
then remove it from the fire, and in twenty-four liours decant the 
clear oil, which should stand for a i'aw weeks before it is used for 
varnisli. 



88 




Steam Power and the Steam-E-> 


^GINE. 










TABLE 








Of the P 


ressure on a square and circular 


Inch, 


respectively. 


exerted 


by the 


elastic 


force of Steam at various degrees of Temperature, with 


the Height of 


the column of Mercury it will support. 




1. PRESSURE ON A SQUARE INCH. 


I. PRESSURE ON A CIRCULAR INCH. 


n 

ii 




m 


3 

ft 


1^ 


c 
c.S 

o Z 

<o si 
III 


ill 


5 
o 

SI 

II 


o 








o 








220 


H 


1-963 


5-15 


222 


2i 


3-183 


6-56 


222 


3 


2-356 


6-18 


224 


3 


3-819 


7-87 


223 


H 


2-749 


7-21 


226 


H 


4-456 


9-18 


225 


4 


3-141 


8-24 


228 


4 


5-098 


10-5 


227 


H 


3-534 


9-27 


230 


H 


5-729 


11-8 


228 


5 


3-927 


10-3 


232 


5 


6-366 


13-1 


230 


H 


4-320 


]l-3 


234 


oi 


7 002 


14-4 


231 


6 


4-712 


12-3 


235 


6 


7 639 


15-7 


233 


H 


5105 


13-4 


236 


6i 


8-276 


17-0 


231 


1 


5-498 


14-4 


238 


7 


8-912 


18-3 


235 


n 


5-890 


15-4 


239 


n 


9-549 


19-7 


236 


8 


6-283 


16-5 


241 


8 


10-18 


21-0 


237 


H 


6-676 


17-5 


242 


8i 


10-82 


22-3 


239 


9 


7-068 


18-5 


244 


9 


11-45 


23-6 


240 


H 


7-461 


19-6 


245 


n 


12-09 


24-9 


241 


10 


7-854 


20-6 


247 


10 


12-73 


26-2 


242 


lOi 


8-247 


21-6 


248 


lOi 


13-36 


27-5 


243 


11 


8-639 


22-6 


250 


11 


14-00 


28 9 


244 


Hi 


9-032 


23-7 


251 


IH 


14-64 


30-1 


245 


12 


9-424 


24-7 


252 


12 


15-27 


31-5 


252 


15 


11-78 


30-9 


259 


15 


19-09 


39-3 


261 


20 


15-71 


41-2 


270 


20 


25-46 


52-5 


269 


25 


19-63 


51-5 


278 


25 


31-83 


65-6 


276 


30 


23-56 


61-8 


287 


30 


38-19 


78-7 


283 


35 


27-49 


72-1 


294 


35 


44-56 


91-8 


289 


40 


31-41 


82-4 


300 


40 


60-92 


105 


294 


45 


35-34 


92-7 


305 


45 


57-20 


118 


300 


50 


39-27 


103 


309 


50 


63-66 

1 


131 





Amalgams — Yauntshes. 8^ 



AMALGAMS. 

When mercury is alloyed with any metal the compound is 
called an amalgam of that metal ; as, for example, an amalgam of 
tin, bismuth, c^c. 

Amalgam for Electrical Machines. 

1. Fuse 1 oz. of zinc with i oz. of tin, at as low a temperature as 
possible ; then add 1-^ oz. of quicksilver, previously made hot ; 
mix, pour out, and when cold reduce it to powder, and triturate it 
with sufficient quicksilver to bring it to a proper consistence. 

2. Zinc 1 part; tin 1 ; quicksilver 2. Melt together. 

3. Zinc 2 parts ; tin 1 ; mercury 5. 

4. La Beaume's. Pour into a chalked wooden box 6 oz. of quick- 
silver ; put into an iron ladle ^ oz. of beeswax, with 2 oz. of puri- 
fied zinc, and 1 oz. of grain tin ; set it over a brisk fire, and when 
the metals are melted pour them into the box, avoiding the dross. 
When cold reduce it to powder, and mix it with lard. Keep it in 
a box covered with tallow, and spread it on leather for use. 

Liquid Amalgam for Silvering Globes, d:c. 
Pure lead 1 oz ; grain tin 1 oz. ; melt in a clean ladle, and imme- 
diately add 1 oz. of bismuth. Skim off the dross, remove the ladle 
from the fire, and before the metal sets add 10 oz. of quicksilver. 
Stir together, avoiding the fumes. 

Amalgam for Varnishing Plastic figures. 
Melt 2 oz. of tin with -^ oz. of bismuth, and add ^ oz. of quick- 
silver. When cold grind it with white of egg, and apply to the 
figure. 

YARNISHES. 

Preparations of Lac. 

Stick-lac consists of twigs of several kinds of trees encrusted with 
a resinous matter, produced by the puncture of an insect called the 
cocus lacca. This, triturated with water, and dried, forms eeed- 
lac. The seeddac, when heated and pressed in cotton bags, forms 
shell-lac. Lac dye is the coloring matter extracted from stick-lac 
by water, and evaporated to dr3-ness, with the addition of earthy 
matters, and formed into square cakes. Seed-lac and shell-lac are 
chiefly used in varnishes, dissolved in rectified spirits, or rectified 
wood naphtha. The alcoholic solution is rendered paler, so that it 
may be used for polishing light colored woods, by digesting it in 
the sun, or near a fire, for two or three weeks, with good animal 
charcoal, and then filtering it through paper in a funnel heated 
with hot water. Shell-lac may be bleached by dissolving it in a 
solution of potash, or soda, and passing chlorine into the solution. 



90 Vaunishes. 



The precipitated lac is collected, and well washed. Kastner 
directs 3 parts of carbonate of potash to be dissolved in 24 of 
water, and 3 of lime added, and the whole digested in a close vessel 
for twenty-four hours. The clear liquor is poured off, and boiled 
with 4 parts of shell-lac. When cold, dilute with 4 times its bulk 
of water, and filter; then add chloride of lime, and afterwards 
diluted muriatic acid. "With these preliminary remarks we come 
now to the lacquers, or varnishes. 

77ie Famoits Brilliant French Varnish for Boots and Shoes. 
Take f of a pint of spirits of wine ; 5 pints white wine ; ^ pound 
of powdered gum Senegal; 6 oz. loaf sugar; 2 oz. powdered galls; 
4 oz. green copperas. Dissolve the sugar and gum in the wine. 
When dissolved, strain ; then put it on a slow fire, being careful not 
to let it boil. In this state put in the galls, copperas, and thel 
alcohol, stirring it well for five minutes. Then set off, and when ' 
nearl}^ cool strain through flannel, and bottle for use. It is applied; 
with a pencil brush. If not sufficiently black a little sulphate of i 
iron, and half a pint of a strong decoction of logwood, maybe 
added, with ^ oz. pearlash. j 

Black Varnish. ,j 

Take any varnish, of the class you wish, 16 parts; lampblack 2! 
parts. Grind the black in a small quantity of the varnish, then'; 
mix it with the remainder. 

Cabinet-maker^ Varnish. 
Pale shell-lac 700 parts ; mastic 65 parts ; strongest alcohol 1000 
parts. Dissolve. Dilute with alcohol. 

Callotfs Soft Etching Var7iish. 
Linseed oil 8 parts ; benzoin 1 part ; white wax 1 part. Melt 
and keep it heated until reduced to two thirds. 

Pale Carriage Varnish. 

Copal 32 parts ; pale oil 80 parts. Fuse and boil until stringy ; 
then add dried white copperas 1 part ; litharge 1 part Boil again, 
then cool a little, and mix in spirits of turpentine 150 parts. Strain. 
While making the foregoing, take of gum anime 32 parts; pale oil 
80 parts ; dried sugar of lead 1 part ; litharge 1 part ; spirits of 
turpentine 170 parts. Pursue the same treatment as before, and 
mix the two compositions while hot. 

Second Quality of Carriage Varnish. 
Take of gum anime 32 parts ; oil 100 parts ; spirits of turpentine 
150 parts; litharge 1 part; dried sugar of lead 1 part; dried cop- 
peras 1 part. Proceed as above. 

Copal Varnish. 
Copal 30 parts ; drying oil 25 parts ; spirits of turpentine 50 
parts. Put the copal into a vessel capable of holding 200 parts, 



Varnishes. 91 



and fuse it as quickly as possible, then add the oil, previously ' 
heated to nearly the boiling point. Mix well, then cool a little, 
and add the spirit of turpentine; again mix well, and cover up, 
until the temperature has fallen to 140° Fah. ; then strain. 

To Dissolve Copal in Spirit. i 

Take the copal and expose it in a vessel formed like a colander : 
to the front of a fire, and receive the drops of melted gum in a ' 
basin of cold water ; then well dry them, in a temperature of about 
95° Fah. By treating copal in this way it acquires the property 
of dissolving in alcohol. • 

Black Copal Varnish. 

Take lamp-black, or ivory-black, in fine powder, and mix it v/itli 

the varnish. j 

Blue Copal Varnish, ; 

Indigo, Prussian blue, blue verditer, or ultra-marine. These ; 

substances must be powdered fine. Proceed as before. j 

Fine Pale Copal Varnish. 
Pale African copal 1 part. Fuse, then add hot pale oil 2 parts. 
Boil -until the mixture is stringy, then cool a little, and add 3 2:>ai'ts ; 
of pale spirits of turpentine. Mix well. | 

Flaxen Grey Copal Vainiish. j 

Ceruse, which forms the ground of the paste, mixed with a small i 

quantity of Cologne earth, as much English red, or carminated j 

lake, and a particle of Prussian blue, and color the varnish there- ' 

with. I 

Green Copal Varnish. \ 

Verdigris, crystallized verdigris, compound green (a mixture of: 

yellow and blue). The first two require a mixture of white in I 

proper proportions, from a fourth to two-thirds, according to the ! 

tint intended to be given. The white used for this purpose is ! 

ceruse, or the white oxide of lead, or Spanish white. Proceed as j 

before. j 

Improved Copal Varnish. \ 

Caoutchoucine (white and scentless), strong alcohol, equal parts;! 

copal in the proportion of two pounds to a gallon. Digest in a i 

close vessel, without heat, for one week. i 

Pearl Grey Copal Varnish. 
White and black; white and blue; for example, ceruse and 
lamp-black; ceruse and indigo. Mix them with the varnish, 
according to the tint recpiired. 

Purple Copal Varnish. 
Prussian blue and vermilion, or any other blue and red ; then 
proceed as before. i 



92 Varnishes. 



Red Copal Varnish. 

1. Vermilion, red oxide of lead (minium), red ochre, or Prussian 
red, <fcc., and proceed as before. 

2. Dragon's blood, brick red, or Venetian red, <fec., and proceed 
as before. 

Violet Copal Varnish. 
Vermilion, blue, white, in proportions required to color the 
varnish. 

White Copal Varnish. 
Copal 16 parts; melt, and add hot linseed oil 8 parts; spirits of 
turpentine 15 parts; finest white lead to color. 

Yellow Copal Varnish. 
Yellow oxide of lead, or IS^aples and Montpelier, both reduced to 
impalpable powder. These yellows are hurt by contact with iron or 
steel. In mixing them, therefore, a horn spatula, with a glass 
mortar and pestle, must be employed. Or gum gutta?, yellow 
ochre, or Dutch pink, according to the nature and tone of the color 
to be imitated, and proceed as before. 

Mastic Varnish. 
Gum mastic 5 pounds ; spirits of turpentine 2 gallons. Mix with 
a moderate heat (carefully applied), in a close vessel, then add 
pale turpentine varnish 8 pints. Mix well. 

Another, 
Mastic 1 pound ; white wax 1 ounce ; oil of turpentine 1 gallon. 
Reduce the wax and mastic small, then digest in a close vessel, with 
heat, until dissolved. 

Common Oil Varnish. 
Resin 4 pounds; genuine beeswax i pound; boiled oil 1 gallon. 
Mix with heat, then add spirits of turpentine 2 quarts. 

Turpentine Varnish. 
Resin 1 part ; boiled oil 1 part. Melt, then add turpentine 2 
parts. Mix well. 

White Hard Spirit Varmsh. 
Gum sandarach 2^ pounds; alcohol (65 op.) 1 gallon. Place 
them in a strong, well closed vessel, and apply the heat of warm 
water, with occasional agitation, until dissolved ; then add pale 
turpentine varnish 1 pint. Mix well, and let the whole rest for 
twent3'-four hours, when it will be ready for use. 

White Spirit Varnish. 
Strongest alcohol 100 parts; sandarach 25 parts; tears mastic 6 
parts ; elemi 3 parts ; Venice turpentine 3 parts. Dissolve in a 
closely corked vessel. 



Varnishes. 



Varriish for Toys. 
Copal 7 parts ; mastic 1 part ; Venice turpentine -^ part ; strongest 
alcohol 11 parts. Dissolve the copal first, with the aid of a little 
camphor, then add the mastic, tfec, and thin with alcohol, as 
required. 

To Clean Varnhh. 
Use a ley of potash, or soda, mixed with a little powdered 
chalk. Do not make the liquor too strong of the alkaU. 

To PoHsh Varnish. 
Take 2 oz. powdered tripoli, put it in an earthen pot, witli 
water to cover it ; then take a piece of white flannel, lay it over a 
piece of cork or rubber, and proceed to polish the varnish, always 
wetting it with the tripoli and water. It will be known when the 
process is finished by wiping a part of the work with a sponge, and 
observing whether there is a fair even gloss. When this is the 
case, take a bit of mutton suet and fine flour, and clean the woi-k. 

Varnish for Harness. 
Take ^ pound of India-rubber ; one gallon of spirit of turpen- 
tine; dissolve enough to make it into a jelly; then take equal 
quantities of good hot linseed oil, and the above mixture. Incor- 
porate them well on a slow fire, and it is fit for use. 

A Varnuhfor Fastening the Leather on Top Rollers in Factories. 
Dissolve 2f oz. of gum arable in water ; and a like amount of 
isinglass dissolved in brandy, and it is fit for use. 

A VarnisJi to Preserve Glass from the Rays of the Sun. 
Reduce a quantity of gum tragacanth to fine poAvder, and let it 
dissolve for twenty-four hours in white of eggs well beat up ; then 
rub it gently on the glass with a brush. 

A fine Black Varnish for Coaches and Iron Work. 
Bitumen of Palestine 2 oz. ; I'esin 2 oz. ; umber 12 oz. Melt 
them separately, and then mix together over a moderate fire. Then 
pour upon them, while on the fire, 6 oz. clear boiled linseed oil, 
stirring the whole from time to time. Take it oft' the fire, and 
when moderately cool pour in 12 oz. of essence of turpentine. 

Varnish for Clock Faces. 
Spirits of wine 1 pint; divide it into four parts; mix one part 
with ^ an oz. of gum mastic in a bottle by itself; one part of 
spirit and ^ oz. gum sandarach in another bottle ; and one part spirit 
and i oz. whitest part of gum benzoin. Mix and temper them to 
suit; if too thick add spirit; if too thin a little mastic; if too soft 
some sandarach or benzoin. When about to use it warm the sil- 
vered plate before the fire, and with a flat camel-hair pencil stroke 
it over till no white streaks appear; tliis will preserve it for many 
years. 



94 Varnishes. 



Brovm Varnish. 

Rectified spirit 2 gallons ; sandarach 3 pounds ; shell-lac 2 pounds ; 
pale turpentine varnish 1 quart. Put them into a tin bottle, cork 
securely, and agitate frequently, placing the tin occasionally in hot 
water till the gum is dissolved, then add a quart of pale turpentine 
varnish. 

Brilliant Amber Spirit Varnish. 

Fused amber 4 oz. ; sandarach 4 oz. ; mastic 4 oz. ; highly rec- 
tified spirit 1 quart. Expose to the heat of a sand bath, with 
occasional agitation, till dissolved. The amber is fused in a close 
copper vessel, having a funnel-shaped projection, which passes 
through the bottom of the furnace by which the vessel is heated. 

Chinese Varvish. 
Mastic 2 oz. ; sandarach 2 oz. ; rectified spirit 1 pint. Close the 
matrass with bladder, with a pin hole for the escape of vapor; 
heat to boiling in a sand or water bath, and when dissolved strain 
through linen. 

Crystal Varnish. 
Picked mastic 4 oz. ; rectified spirit 1 pint; animal charcoal 1 
oz. Digest, and filter. 

Picture Varnish. 
Ohio turpentine 2 oz. ; mastic 12 oz,; camphor ^ drachm; pounded 
glass 4 oz. ; rectified oil of turpentine 3 pints. This is for oil 
paintings. 

Canada Varnish. 
Clear balsam of Canada 4 oz. ; camphene 8 oz. Warm gently, 
and shake together till dissolved. This varnish is for maps, draw- 
ings, <fec., which must be first sized over with a solution of isinglass, 
taking care that every part is covered. When dry, the varnish is 
brushed over it, 

Tingrifs Essence Varnish. 
Powdered mastic 12 oz. ; pure turpentine 1^ oz. ; camphor ^ oz. ; 
powdered glass 5 oz. ; rectified oil of turpentine 1 quart. 

Common Turpentine Varnish. ^ 

This is merely clear pale resin, dissolved in oil of turpentine; 
usually 5 pounds of resin to 7 pounds of turpentine. 
Amber Variiish. 
Amber 16 oz. ; melt in an iron pot, and add ^ pint of drying 
linseed oil, boiling hot, and add 3 oz. resin, and 3 oz. asphalte, each 
in fine powder. Stir till they are thoroughly incorporated ; 
remove from the fire, and add a pint of warm oil of turpentine. 
Balloon Varnish. 
Melt india-rubber in small pieces with its weight of boiled 
linseed oil, and thin it with oil of turpentine. 



Varnishes. 95 



Varnish for Engraving on Copper, 
YeJloAV wax 1 oz. ; mastic 1 oz. ; asphaltum ^ oz. Melt, pour 
into water, and form into balls for use. A softer varnish for 
engravers is made thus: Tallow 1 part, and 2 of yellow wax; or, 
with 2 oz. wax, 1 drachm common turpentine, and 1 drachm olive 
oil. 

Etching Varnishes. 
White Wax 2 oz. ; asphaltum 2 oz. Melt the wax in a clean 
pipkin, add the asphaltum in powder, and boil to a proper con- 
sistence. Pour it into warm water, and form it into balls, which 
must be kneaded, and put into taffeta for use. 
Another. 
"White wax 2 oz. ; Burgundy pitch ^ oz ; black pitch -i- oz, ; melt 
together, and add by degrees 2 oz. powdered asphaltum, and boil 
it till a drop cooled on a plate becomes brittle. 
Another. 
Equal quantities of linseed oil and mastic, melted together. 

Engraving Mixture for Writing on Steel. 
Sulphate of copper 1 oz. ; sal ammoniac -t oz. Pul\^erize sepa- 
rately, adding a little vermilion to color it, and mix with 1^ oz. ' 
vinegar. Rub the steel with soft soap, and write with a hard, 
clean pen, without a slit, dipped in the mixture. j 

Etching Fluids. ' 

For Copper. — 1. Aquafortis 2 oz. ; water 5 oz. Mix. 
2. Callot's Eau Forte for Fine Touches. — Dissolve 4 parts each of 
verdigris, alum, sea salt, and sal ammoniac, in 8 parts vinegar; add 
16 parts water, boil for a minute, and let it cool. 

Yov Steel.— 1. Iodine 1 oz. ; iron filings ^ drachm ; water 4 oz. 
Digest till the iron is dissolved. 

2, Pyroligneous acid 4 parts by measure ; alcohol 1 part. Mix, 
and add 1 part double aquafortis (sp. gr. 1-28). Apply it from H 
to 15 minutes. 

Varnish for Engraving on Glass. 
Wax 1 oz. ; mastic \ oz. ; asphaltum \ oz. ; turpentine \ drachm. 

Another. 
Mastic 1 5 parts ; turpentine 7 ; oil of spike 4. 

Le Blond's Varnish. 
Keep 4 pounds balsam of copaiva warm in a sand or water bath, 
and add 16 oz. of copal, previously fused and coarsely powdered, 
by single ounces, daily, and stir it frequently. When dissolved add 
a little Ohio turpentine. 

Sealing Wax Varnish. 
Black or colored sealing wax, broken small, and sufficient recti- 
fied spirit to cover it ; digest till dissolved. I 
Black Japan. 
Boil together a gaUon of boiled linseed oil, 8 oz. imiber, and 8 
oz. asphaltum. When sufficiently cool thin it with oil of turpentine. 



96 Varnishes. 



Brunswick Black. 
Melt 4 pounds asphaltum, add 2 pounds hot linseed oil, and when 
sufficiently cool add 1 gallon oil of turpentine. 

Varnish for Gun Barrels, after browning them. 
Shell-lac 1 oz ; dragon's blood ^ oz. ; rectified spirit 1 quart. 
Dissolve and filter. 

Transfer Varnisli. 
Alcohol 5 oz. ; pure Yenice turpentine 4 oz. ; mastic 1 oz. 

Hair Varnish. 
Dissolve 1 part of clippings of pigs' bristles, or horsehair, in 10 
parts of drying linseed oil, by heat Fibrous materials (cotton, 
flax, silk, (fee), imbued with the varnish and dried, are used as a 
substitute for hair cloth. 

Glass Varnish. 
This is a solution of soluble glass, and is thus made: Fuse 
t02;ether ]5 parts powdered quartz (or fine sand), 10 parts potash, 
and 1 charcoal. Pulverize the mass, and expose it for some days to 
the air; treat the whole with cold water, which removes the 
foreign salts, <fec. ; boil the residue in 5 parts of water imtil it 
dissolves. It is permanent in the air, and not dissolved by water. 
This varnish is used to protect wood, <fec., from fire. 

Varnish for Gilded Articles. 

Gum-lac 4 parts; dragon's blood 4; annatto 4; gamboge 4; 
safiron 1. Dissolve each resin separately in 8 parts alcohol, and I 
make a separate tincture with the dragon's blood and annatto, also 
in 8 parts alcohol each; then mix the former together, and add a 
sufficient quantity of the tinctures to give the required shade and ! 
color to the varnish. 

Gold Varnishes. 

Turmeric 1 drachm ; gamboge 1 drachm ; oil of turpentine 2 
pints; shell-lac 5 ounces; sandarach 5 oz. ; dragon's blood 7 
drachms ; thin mastic varnish 8 oz. Digest, with occasional agita- 
tion, for fourteen days, in a warm place ; then set it aside to fine, 
and pour off the clear. 

Another. 

Dutch leaf 1 part; gamboge 4; gum dragon 4; proof spirit 18. 
Macerate for twelve hours, then grind on a stone slab. 

Varnish for Water Color Drawings. 
Canada balsam 1 pint; oil of turpentine 2 parts, mixed. Size 
the drawing before applying the varnish. 

Earthenware Varnish. 
Flint glass 1 part ; soda 1. Mix. 

Magilp. 
Mastic varnish 1 part; drying oil 2. Mix. 



Practical Tables. 



97 



Mix. 



Mix. 



Another, 
Mastic varnish 1 part; drying oil 1. 
Another. 
Equal parts of mastic varnish, drying oil, and turpentine. 

Metallic VarnUh for Coach Work, (Jcc, 
Asphaltum 56 pounds. Melt, then add litharge 9 pounds; red 
lead 7 pounds; boil, then add boiled oil 12 gallons; yellow resin 
12 pounds. Again boil, until in cooling the mixture may be rolled 
into pills; then add spirit of turpentine 30 gallons; lampblack 7 
pounds. Mix well. 

Impermeahle Varnish. 
Boiled oil 100 parts; finely powdered litharge 6 parts ; genuine 
beeswax 5 parts. Boil until sufficiently thick and stringy, then 
pour off the clear. 

Engravers^ Stopping-out Varnish. 
Take lampblack and turpentine to make a paste. 



PEAOTICAL TABLES. 

I Weight of Metals— Wrought Iron ; Square, Round, and Flat. 

] Table 1. contains the weight of Square Iron in sizes, from ^ inch 
'. to six inches square, advancing by ^ inch; and from 6 to 12 inches 
square, advancing by i inch ; and in lengths, from 1 foot to 18 feet. 
The sizes are arranged in the first column of each page, and the 
; lengths along the top ; the weight in lbs. immediately under the 
, lengths, and in a line with the sizes. j 

i Table 11. contains the weight of Round Iron in sizes from ^ inch ' 
i to 6 inches diameter, advancing by ^ inch ; and from 6 to 12 inches 
diameter, advancing by J inch; and in lengths, from 1 foot to 18! 
feet. The sizes, lengths, and weights are arranged as in Table I. | 
Table III. contains the weight of Flat Iron in widths, from J inch [ 
to 6 inches diameter, advancing by ^ inch ; in thicknesses, from 
i inch to 1 inch, advancing by ^ inch ; and in lengths, from 1 
to 18 feet. The widths, lengths, and v/eights are arranged as in 
the preceding tables, and the thicknesses alongside of the widths. 

The tables are all calculated to the nearest tenth of a pound. 
To the weights of bars of Wrought Iron add y^ yth part for bars of 
8oft steel ; and from the same weights subtract y\t\i part for bars 
of Oast Iron. 



98 



Practical Tables. 



TABLE I. 

SQUARE IRON. 



Size. 


1 ft. 


2 ft. 


3 ft. 


4 ft. 


6 ft. 


6 ft. 


7 ft. 


8 ffc. 


9 ft. 


Inch. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


Iba. 


lbs. 


lbs. 


lbs. 


i 


0-2 


0-4 


0-6 


08 


1-1 


1-3 


1-5 


1-7 


1-9 


t 


0-5 


1-0 


1-4 


1-9 


2-4 


2-9 


3-3 


3-8 


4-3 


i 


0-8 


1-7 


2-5 


3-4 


4-2 


5-1 


5-9 


6-8 


7-6 


f 


1-3 


2-6 


4-0 


5-3 


6-6 


7-9 


9-2 


10-6 


11-9 


f 


1-9 


3-8 


5-7 


7-6 


9-5 


11-4 


13-3 


15-2 


17-1 


i 


2-6 


5-2 


7-8 


104 


12-9 


15-5 


18-1 


20-7 


23-3 




3-4 


68 


10-1 


13-5 


16-9 


20-3 


23-7 


27-0 


30-4 


H 


4*3 


8-6 


12-8 


17-1 


21-4 


26-7 


29-9 


34-2 


38-5 


li 


5-3 


10-6 


15 8 


21-] 


264 


31-7 


37-0 


42-2 


47-5 


If 


6-4 


12-8 


19-2 


25-6 


32-0 


38-3 


44-7 


51-1 


57-5 


H 


7-6 


15-2 


22-8 


30-4 


38-0 


45-6 


53-2 


60-8 


68-4 


If 


8-9 


17-9 


26-8 


35-7 


44-6 


53-6 


62 6 


71-4 


80'3 


If 


10-4 


20-7 


31-1 


41-4 


51-8 


621 


72-0 


82-8 


93-2 


n 


119 


23-8 


35-6 


47-0 


59-4 


71-3 


83-2 


95-1 


106-9 


2 


13-5 


27-0 


40-6 


54-1 


67-6 


81-1 


94-6 


108-2 


121-7 


2i 


15-3 


30-5 


45-8 


61-1 


76-3 


91-6 


106-8 


122-1 


137-4 


2i 


17-1 


34-2 


51-3 


68-4 


85-6 


102-7 


119-8 


136-9 


154-0 


2f 


19-1 


38-1 


57-2 


76-3 


95-3 


114-4 


133-5 


152-5 


171-6 


2i 


2M 


42-2 


634 


84-5 


105-6 


126-7 


147-8 


169-0 


190-1 


2f 


23-3 


46-6 


69-9 


93-2 


116-5 


139-8 


163-0 


186-3 


209-6 


2f 


25-6 


5M 


76-7 


102-2 


127-8 


153-4 


178-9 


204-5 


230-0 


2J 


27-9 


55-9 


83-8 


1118 


139-7 


167-6 


195-7 


223-5 


251-5 


3 


30-4 


60-8 


91-2 


121-7 


152-1 


182-5 


212-9 


243-3 


273-7 


34 


33-0 


66-0 


99-0 


132-0 


165-1 


198-1 


231-1 


264-1 


297-1 


H 


35-7 


71-4 


107-1 


142-8 


178-5 


214-2 


249-9 


285-6 


321-3 


H 


38-5 


77-0 


115-5 


154-0 


192-5 


281-0 


269-5 


308-0 


346-5 


H 


41-4 


82-8 


124-2 


165-6 


207-0 


248-4 


289-8 


331-3 


372-7 


3f 


44-4 


88-8 


133-3 


177-7 


222-1 


266-5 


310-Vt 


355-3 


399-8 


8f 


47-5 


95-1 


142-6 


190-1 


237-7 


285-2 


332-7 


380-3 


427-8 


n 


50-8 


101-5 


152-3 


203-0 


253-8 


304-5 


355-3 


406-0 


456-8 



Practical Tables. 



99 





TABLE L 




s 


QU a RE IRON. 


Size 


. 10 ft 


11 ft 


12 ft 


. 13 ft. 14 ft. 


15 ft. 


16 ft. 


17 ft 


18 ft. 


Inch. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


. lbs. 


lbs. 


i 


2-] 


2-g 


2-. 


) 2'' 


1 3-C 


) 3-2 


3-^ 


1 3-6 


3-8 i 


f 


4-^ 


5-2 


5-- 


J 6-^ 


I 6-7 


7-1 


7-6 8-1 


8-6! 


i 


8-5 


9-3 


10-] 


ll-( 


) 11-8 


12-0 


13-5 14-4 


15-2! 


f 


13-2 


14-5 


15-^ 


17-1 


I 18-5 


19-8 


21-1 


I 22-4 


23 8 1 


f 


19-0 


20-9 


22-8 


24-/ 


26-6 


28-5 


30 --^ 


[ 82-3 


34-2 1 


i 


25-9 


28-5 


31-1 


33-6 


36-2 


38-8 


41-^ 


44-0 


46-6! 


1 


33-8 


37-2 


40-6 


43-9 


47-3 


50-7 


54-1 


57-5 


60-8 


H 


42-8 


47-1 51-3 


55-6 


59-9 


64-2 


68-4 


72-7 


77-0 1 


n 


52-8 


58-1 


63 4 


68 6 


73-9 


79-2 


84-5 


89-8 


95-0! 


If 


63-9 


70 3 


76-7 


83-1 


89-5 


95-9 


102-2 


108-6 


115-0 ; 


u 


16-0 


83-6 


91-2 


95-9 


106-5 


114-1 


121-7 


129-3 


136-9 


If 


89-3 


98-2 


107-1 


116-0 


125 


133-9 


142-8 


151-7 


160-7 


If 


103-5 


133-9 


124-2 


134-6 


144-9 


1553 


165-6 


176-0 


186-3, 


n 


118-8 


130-7 


142-6 


154-5 


166-4 


178-2 


1901 


202-0 


213-9 


2 


135-2 


148-7 


162-2 


175-8 


189-3 


202-8 


216-8 


229-8 


243-4: 


H 


152-6 


167-9 


183-2 


198-4 


213-7 


228-9 


244-2 


259 5 


274-7 1 


H 


nil 


188-2 


205-3 


222-5 


239-6 


256-7 


273-8 


290-9 


308-0 ' 


H 


190-7 


209-7 


228-8' 


217-9 


266-9 


286-0 


305-J 


324-1 


843-2: 


H 


211-2 


232-3 


253-4' 


274-6 


295-7 


316-8 


337 -e 


359-0 


880-2 : 


H 


232-9 


256-2 


279-5, 


302-8 


326-1 


349-4 872-7 


8960 


419-3! 


2f 


2556 


281-2 


306-7 


332-3 


357-8 


383-4 409-0 


484-5 


460-1 


2| 


279-4 


307-3 


335-3 


363-2 


391-1 


419-1 


447-0 


475-0 


502-9 


3 


304-2 


334-6 


365-0 


395-4 


425-8 


4562 


486-7 


517-1 


547-5! 


8i 


3301 


363-1 


3961 


4291 


4621 


495-2 


528-2 


661-2 


594-2 1 


3i 


357 


392-7 


428-4 


464-2 


499-9 


535-6 


571-3 


607-0 


642-7 1 


H 


385-0 


423-5 


462-0 


500-5 


539-0 


577-5 


616-0 


654-6 


693-1 ;' 


H 


414-1 


455-5 


496-9 


638 3 


579-7 


621-1 


662-5 


703-9 


745-3! 


n 


444-2 


488-6 


533-0 


577-4 


621-9 


666-3 


710-7 


755-1 


799-5 


3f 


475-3 


522-9 


570-4 


617-9 


665-5 


713-0 


760-c 


808-1 


855-6 


3J 


507-6 


558-3 


609-1 


669-8 


710-6 


761-3 


812-1 


862-9 


913-6 



100 






Practical Tables. 












TABLE I. 








8 Q U A p. E IRON. 




Size. 


1 ft. 


2 ft. 


3 ft. 


4 ft. 


5 ft. 


6 ft. ■ 


7 ft. 


8 ft. 


9 ft. : 

1 


Inch. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


i 

lbs. 1 


4 


54-1 


108-2 


162-8 


216-3 


270-4 


324-5 


378-6 


432-7 


486-8 


H 


57-5 


1150 


172-6 


230-1 


287-6 


845 1 


402-6 


460-1 


517-7 


4i 


61-1 


122-1 


183-2 


244-2 


305 -S 


366-3 


4-27-4 


488-4 


549-5 


4| 


64-7 


129-4 


194-1 


258-8 


323-0 


3b8-2 


452-9 


517-6 


582-3 


U 


68-4 


U6-{- 


205 8 


273-8 


342-2 


410-7 


479-1 


547-6 


616-0 


4| 


72-3 


144-e 


216 t 


289-2 


361-5 


433-8 


506-1 


578-4 


650-7 


4f 


76-3 


152-0 


228-S 


3051 


381-3 


457-6 


533-8 


610-1 


686-4 


4J 


80-3 


160-7 


241 -U 


321-3 


401-7 


482-U 


562-3 


642-7 


728-0 


5 


84*5 


169-0 


253-4 


337-9 


422-4 


506-9 


591-4 


67 5 -f 


760-3 i 


H 


88-8 


177-6 


266-4 


355-1 


443 -y 


532-7 


621-5 


710-3 


799-1 i 


^i 


93-2 


186-S 


279-0 


3727 


4658 


559-0 


652-2 


745-3 


838-51 


5f 


97-7 


195-:^ 


293-0 


390-6 


488-3 


585-9 


683-6 


781-8 


878-9! 


H 


102-2 


204-5 


306-7 


409-0 


511-2 


613-4 


715-7 


817-e 


920-2; 


5f 


107-0 


213 5 


320-9 


427-8 


534-8 


641-7 


748-7 


855-e 


962-6! 


5i 


111-8 


223-5 


335-3 


447-0 


558-8 


670-5 


782-3 


894-0 


1005-8! 


51 


116^7 


233-3 


350-0 


466-7 


583-4 


7U0-0 


816-7 


933-4 


1050-0 


6 


121-7 


243-8 


365-0 


486-7 


608-3 


730-0 


841-6 


973-8 


1095-0 


6i 


132-0 


264-1 


396-1 


528-2 


660-ii 


792-2 


924-31056-8 


1188-41 


6i 


142-8 


285 -c. 


428-4 


571-3 


714-1 


856-9 


999-7 1142-0 


1285-3 ! 


6J 


1540 


30S-0 


462-0 


616-0 


770-1 


924-lil078-l 

1 


:i232-l 


1386-1 i 


7 


165-6 


331-2 


496-9 


662-5 


828-2 


j 
993-8:1159-4 


1325-1 


1490-7 


Vi 


177-7 


355-3 


533-0 


710-7 


888-4 


1066-0 1243-7 1421-4 


1 599-0 i 


'^i 


190-1 


380-3 


570-4 


760-5 


950-7 


1140-8:1331-01521-1 


1711-2 


Vl 


203-0 


406-0 


609-1 


812-1 


1015-1 


1218-1 1421-2 

1 


1624-2 


1827-2 


8 


216-3 


432-7 


649-0 


865-3 


1081-7 


1298-0 


1514-4 


1730-7 


1947-0 


8^ 


230-1 


460-1 


690-2 


920-3 


1150-3 


1380-4 


1610-5 


1840-5 


2070-6 


8i 


244-2 


488 4 


732-7 


976-9 


1221-1 


1465-3 


1709-5 


1958-8 


2198-0 


81 


258-8 


517-6 


776-4 


1035-2 


1294-0 


1552-8 


1811-6 


2070-4 


2329-2 


9 


273-8 


547 6 


821-4 


1095-2 

1 


13S9-0 


1642-8 


191b-5 


2190-3 


2564-1 









Practical Tables. 






101 


TABLE I. 


SQUARE IRON. 


Size. 


10 ft. 


11 ft. 


12 ft. 


13 ft. 


14 ft. 


15 ft. 


16 ft. 


17 ft. 


18 ft. 


Inch. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


4 


540-8 


594-9 


649-0 


703-1 


7571 


811-3 


865-3 


919-4 


973-51 


4i 


6lo-2 


632-7 


690-2 


747-7 


805-2 


862-8 


920-3 


977-8 


1035-3: 


4i 


GlO-6 


671-6 


732-7 


793-7 


854-8 


915-8 


976-9 


1037-9 


1099-0 


41 


646-0 


711-i 


776-4 


841-1 


905-8 


970-5 


1035-2 


1099-9 


1164-6' 


4i 


684 5 


752-9 


821-4 


889-8 


958 3 


1026-7 


1095-2 


1163-6 


1232-1 


H 


723-1 


795-4 


8t;7-7 


940-0 


1012-3 


1084-6 


1156-9 


1229-2 


1301-5; 


4f 


762-6 


838-9 


915-2 


991-4 


1067-7 


1144-0 


1220-2 


1296-5|l372-8 


^ 


803-3 


883-7 


9640 


1044-3 


1124-7 


1205-0 


1285-3 


1365-7 


1446-0; 


5 


844-8 


929-3 


1013-8 


1098-2 


11827 


1267-2 


1351-7 


1436-2 


1520-6' 


5i 


887-8 


976-6 


1065-4 


1154-2 


1243-(; 


1331-8 


1420-5 


1509-3 


1598-1^ 


H 


1 931-7 


1024-8 


1118-0 


1211-2 


1304-4 


1397-5 


1490-7 


1583-9 


1677-0, 


51 


976-6 


1074-2 


1171-9 


1269-5 


1367-2 


1464-9 


1662-0 


1660-2 


1757-8' 


6i 


1022-4 


1124-6 


1226-9 


1329-1 


1431-4 


1533-6 


1635-8 


1738-1 


1840-3 : 


5i 


1069-5 


1176-5 


1283-4 


1390-4 


1497-3 


1604-3 


1711-2 


1818-2 


1925-2 


5i 


1117-6 


1229-3 


13411 


1452-8 


1564-6 


1676-3 


1788-1 


1899-9 


2011-6 


H 


1160-0 


1283-4 


1400-1 


1516-7 


1633-4 


1750-1 


1866-7 


1983-4 


2100-1 


6 


1229-6 


1338-3 


14600 


1581-6 


1703-3 


1825-0 


1946-6 


•2068-3 


2190-0 


6i 


1320-4 


1452-4 


1584-4 


1716-5 


1848-6 


1980-6 


2112-6 


2244-7 


2376-7 1 


6i 


1428-2 


1571-0 


1713-8 


1856-6 


1999-4 


2142-2 


2285-1 


•2427-9 


2570-7 \ 


6i 


1540-1 


1694-1 


1848-1 


2002-2 


2056-2 


2310-2 


2464-2 


2618-2 


2112-2, 


7 


1656-3 


1822-0 


1987-6 


2153-2 


2318-8 


2484-5 


2650-1 


2815-7 


2981-4 


Vi 


1776-7 


1954-4 


2132-1 


2309-7 


•2487-4 


2665-1 


2842-8 


3020*4 


3198-1 


n 


1901-4 


2091-5 


2281-6 


•2471-8 


2661-9 


2852-0 


3042-2 


3232-3 


3422-4 


Vi 


2030-2 


2233-3 


2436-3 


2639-3 


2842-3 


3045-4 


3248-4 


3451-4 


3654-4 


8 


2163-4 


2379-7 


2596-0 


2812-4 


3028-7 


3245-0 


3461-4 


3677-7 


3894-0 


H 


2300-7 


2530-7 


•2760-8 


2990-9 


3220-9 


3451 


3681-1 


3911-1 


4141-2 i 


H 


2442-2 


2686-4 


2930-6 


3174-9 


3419-1 


3663-3 


3907-5 


4151-7 


4396-0! 


8| 


2588-0 


2846-8 


3105 6 


3364-4 


3623-2 


3882-0 


4140-8 


4399-6 


4658-4: 


9 


2737-9 


3011-7 


3285-5 


3659-3 


3833-1 


4106-9 


4380-7 


4654-5 


4928-3 



9* 



102 



Practical Tables. 



TABLE I. 



SQUARE IRON. 



Size. 



u 
n 



10 

lOii- 
10| 



11 

Hi 

Hi 
iif 



12 



1 ft. 



2 ft. 



lbs. 

289-2 
305-1 
321-3 



337-9 
350-1 

3Y2 
390-6 



409-0 
427-8 
447-0 
466-7 



3 ft. I 4 ft. 5 ft. I 6 ft. 7 ft. 8 ft 



lbs. 



lbs. 



578-4 867-7;1156-9 
610-1' 915-2^1220-2 
642-7i 964-0:128o-3 

J I 

675-8ll013-8jl351-' 
710-2|1065-4:1420-5 
745-3 1118-0jl490-7 
781-3 1171-9 1562-5 



817-9 1226-9il635-8 
855-6:1283-4il711-2 
894-0!l341-l'l788-l 
933-4 ;1400-i;i866-7 



9 ft. 



lbs. 



lbs. 



lbs. 



lbs. 



lbs. 



1446-1 1735-3 2024-5!2313-8 2603-0 
1525-3 1830-3 2135-4 2440-4 2745-5 
1606-7 1928-0 2249-32570-7 2892-3 



1689-6 2027-5 2365-4;2703-4 3041-0 
1775-7 2130-8 2486-0'2841-l 3196*2 



1863-4 2236-0 2608-7 
1953-112343-8 2734-4 



2981-4 3354-0 
3125-0 3515 7 : 



•2044-8 2453-8 2862-7 3271-7 
2139-1 2566-9 2994*7 3422*5 
2235-1 2682-1 3129-2'3576-2 
2333-4 2800-13266*8 3783*5 



486*7 973-3 1460-0 1946-6 



2433*3 



2919-9 3406-6 



3893*2 



3680*6 
3850-3 
4023-2 
4200-2 



4379-9 



Glazes. — Common earthenware is glazed with a composition con- 1 
taining lead, on which account it is unfit for many pharmaceutical • 
purposes. The folloAving glaze has been proposed, among others, j 
as a substitute : 100 parts of washed sand, 80 of purified potash, 10^ 
of nitre, and 20 of slaked lime; all Avell mixed, and heated in a! 
blacklead crucible, in a reverberatory furnace, till the mass flows i 
into a clear glass. It is then to be reduced to powder. The goods 
to be slightly burnt, placed under water, and sprinkled with the ' 
powder. | 

Glaze for Porcelain. — Feldspar 27 parts, borax 18, Lynn eand 
4, nitre 3, soda 3, Cornwall china clay 3 parts. Melt together to 
form a frit, and reduce it to a powder, with 3 parts of calcined 
borax. 

Solvent for Old Putty and Paint. — Soft soap mixed with solution 
of potash or caustic soda ; or pearlash and slaked lime mixed with | 
sufficient water to form a paste. Either of these laid on with an j 
old brush or rag, and left for some hours, will render it easily 
removable. 



pRACTrcAL Tables. 



1^ 



TABLE I. 

SQUARE IRON. 



Size. 



Inch. 

H 

n 



10 
lOi 
lOi 
10| 



11 
Hi 

in 

111 



12 



10 ft. 



11 ft. 



lbs. 



lbs. 



2892-2 3181-4 
3050-6 3355-6 
3213-3 3534-7 



3379-2 
3551-4 
3726-7 
3906-3 



4089-6 
4278-1 
4470-2 
4666-8 



4866-6 



3717-1 
3906-5 
4099-4 
4297-0 



4498-6 
4705-9 
4917-3 
5133-5 



5353-2 



12 ft. 



lbs. 

3470-6 
3660-7 
3856-4 



4055-0 
4261-6 
4472-1 
4687-5 



4907-4 
5133-7 
5364-3 
5600-2 



5839-9 



13 ft. 



lbs. 

3759-9 
3965-7 
4177-3 



4393-0 
4616-8 
4844-7 
5078-2 



5316-5 
5561 -fi 
5811-3 
6066-9 



6326-5 



14 ft. 



lbs. 



15 ft. 



lbs. 



4049-1 4338-3 

4270-84575-8 
4498-64820-0 



4730-95068-8 
497 1-9 5327 
5217-4 5590-1 
5468-8 5859-4 



5725-46134-4 

59894'6417*2 
6258-36705-4 
6533-6 7000-3 



6813-27299-8 



16 ft. 



lbs. 

4627-5 

4880-9 
5141-3 



5406-/ 
5682-2 
5962-8 
6250-0 



6543-4 
6845-0 
7152-4 
7466-9 



7786-5 



17 ft. 



lbs. 

4916-7 
51860 
5462-6 



18 ft. 



lbs. 

5206-0 
5491-0 
5784-0 



5744-6 6082-6 
6037-3 6392-4 
6335-4 6708-1 
6644-7 7031-3 



6952-3 7361-3 
7272-8 7700-6 
7599-4 8046-4 
7933-6 8400-3 



8273-2 8759-8 



Scouring Drops for REMovrNO Grease. — 1. Alcohol (pure) 6 oz., 
camphor 2 oz., rectified essence of lemon 8 oz. 

2. Camphene 3 oz., essence of lemon 1 oz. Mix. Some direct 
them to be distilled together. 

3. French. Camphene 8 oz., pm-e alcohol 1 oz., sulphuric ether 1 
oz., essence of lemon 1 dr. 

4. Spirits of wine 1 pint, white soap 3 oz., ox gall 3 oz., essence 
of lemon J oz. 

Balls, Heel. — 1. Melt together 4 oz. of mutton sue!, 1 oz, of 
beeswax, 1 oz, of sweet oil, ^ oz. oil of turpentine, and stir in 1 oz. 
of powdered gum arable, and ^ oz. of fine lampblack. 

2. Beeswax 8 oz., tallow 1 oz., powdered gum 1 oz., lampblack 
q. 8. These are used not merely by the shoemaker, but to copy 
inscriptions, raised patterns, <fec., by rubbing the ball on paper laid i 
over the article to be copied. For copying ancient monumental , 
brasses, a similar compound, colored with bronze powder instead 
of lampblack, is sometimes employed. I 



104 






Practical Tabies. 
















TABLE 11. 










' 






ROUND IRON. 








Size. 


1 ft. 


2 ft. 


8 ft. 


4 ft. 


5 ft. 


6 ft. 


7 ft. 


8 ft. 


9 ft. 


Inch. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


Iba. 


i 


0-2 


O'S 


0-5 


0-7 


0-8 


1-0 


1-2 


1-3 


1-5 


f 


0-4 


0-7 


1-1 


1-5 


1-9 


2-2 


2-6 


3-0 


3-4 


i 


0-7 


1-8 


20 


2-7 


3-3 


4-0 


4-6 


5-3 


6-0 


f 


1-0 


2-1 


31 


4-2 


5-2 


6-3 


7-8 


8-8 


9-4 


f 


1-5 


3-0 


4-5 


6-0 


7-5 


9-0 


10 5 


11-9 


13-4 


1 


2-0 


4-1 


6-1 


8-1 


10-2 


12-2 


14-2 


16-3 


18-3 




2-7 


5-8 


8-0 


10-6 


18-8 


15-9 


18-6 


21-2 


23-9 


H 


8-4 


6-7 


10-1 


13-4 


16-8 


20-2 


28-6 


26-9 


30-2 


li 


4-2 


8-3 


12-5 


16-7 


20-9 


25-0 


29 2 


33-4 


37-5 


If 


5-0 


10-0 


io-l 


20-1 


25-1 


30-1 


35-1 


40-2 


45-2 


H 


6-0 


11-9 


17-9 


23-9 


29-9 


35-8 


41-8 


47-8 


53-7 


if 


7-0 


14-0 


21-0 


28-0 


35-1 


42-1 


49-1 


56-1 


63-1 


If 


8-1 


16-8 


24-4 


32-5 


40-6 


48-8 


56-9 


65-0 


73-2 


n 


9-3 


18-7 


28-0 


87-3 


46-7 


56-0 


65-8 


74-7 


84-0 


• 2 


10-6 


21-1' 


81-8 


42-5 


53-1 


68-7 


74-8 


84-9 


95-5 


2i 


12-0 


24-0 


36-0 


48-0 


59-9 


71-9 


88-9 


95-9 


107-9 


: 2i 


13-5 


26-9 


40-3 


53-8 


67-2 


80-6 


94-1 


107-5 


121-0 


! 2| 


15-0 


30-0 


44-9 


60-0 


74 9 


89-9 


104-8 


119-8 


134-8 


2i 


16-7 


33-4 


50-1 


66-8 


83 4 


100-1 


116-8 


138-5 


150-2 


: 21- 


18-8 


86-6 


54-9 


73-2 


91-5 


109-8 


128-1 


146-3 


164-6 


1 2f 


20-1 


40-2 


60-2 


80-3 


100-4 


1205 


140-5 


160-6 


180-7 


2J 


21-9 


43-9 


65-8 


87 8 


109-7 


181-7 


153-6 


175-6 


197-5 


3 


28-9 


47-8 


71-7 


95-6 


119-4 


148? 


167-2 


191 1 


215 


84 


25 9 


51-9 


77-8 


103 7 


129-6 


155-6 


181-5 


207-4 


288-3 


' 3i 


28-0 


56-1 


84-1 


112-2 


140-2 


168-2 


196-3 


224-8 


253-4 


, H 


30-2 


60-5 


90-7 


121-0 


151-2 


181-4 


211-7 


241 9 


272-2 


H 


32-5 


65-0 


97-5 


130-0 


162-6 


195-1 


227-6 


260-1 


292-6 


H 


84-9 


69-8 


104-7 


139-5 


174-4 


209-3 


244-2 


279-1 


814-0 


3f 


87-3 


74-7 


112'0 


149-3 


186-7 


224-0 


261-3 


298-7 


386-0 


3| 


39-9 


79-7 


119-6 


159-5 


199-3 


239-2 


279-0 


318-9 


358-8 



^^^ 






Practical Tables. 






106 






TABLE IL 

ROUND IRON. 




Size. 


10 ft. 


11 ft. 


12 ft. 


13 ft. 


14 ft. 


15 ft. 


16 ft. 


17 ft 


18 ft. 


Inch. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs.. 


lbs. 


lbs. 


lbs. 


i 


1-7 


1-8 


2-0 


2-1 


2-3 


2 5 


2-6 


2-8 


3-0 


f 


3-7 


4-1 


4-5 


4-8 


5-2 


6-6 


60 


6-3 


6-7 


i 


6-6 


7-3 


8-0 


8-6 


9-3 


99 


10 6 


11-3 


11-9 


1 


10-4 


11-5 


12-5 


13-6 


14-6 


15-6 


16-7 


17-3 


18-8 


f 


14-9 


16-4 


17-9 


19-4 


20-9 


22 4 


23-^ 


25-4 


26-9 


i 


20-3 


22-4 


24-4 


26-4 


28-4 


305 


32-0 


34-5 


36-6 




26-5 


29-2 


31-8 


34 5 


37-2 


89 8 


42-0 


45-1 


47-8 


H 


33-6 


37-0 


40-3 


43-7 


47-0 


60-4 


63-8 


57-1 


605 


li 


41-7 


45-9 


50-1 


54-2 


58-4 


62-6 


66-& 


709 


75-1 


If 


50-2 


552 


60-2 


65-2 


70-3 


76-3 


80 8 


85 3 


90-3 


H 


59-7 


65-7 


71-7 


77-6 


83-6 


89-6 


956 


101-5 


107-6 


If 


70-1 


771 


84-1 


91-1 


98-1 


105-2 


112-'i 


119-2 


1262 


If 


81-3 


89-4 


97-5 


105 7 


113-8 


121-9 


130-( 


138-2 


146-3 


n 


93-3 


102-7 


112-0 


121-8 


130-7 


140-0 


149-8 


158-7 


168-0 


2 


106-2 


116-8 


127-4 


138 


148-6 


159 2 


169-9 


180-5 


192-1 


H 


119-9 


131-9 


143-9 


155-8 


167-8 


179-8 


181i 


193-8 


205-8 


H 


134-4 


147-8 


161-3 


174-7 


188-2 


201-6 


21 5-0 


228-5 


241-9 


2f 


149-8 


164-7 


179-7 


194-7 


209-7 


224-6 


239-6 


264-6 


269-6 


H 


166-9 


183-6 


200-3 


216-9 


233-6 


250-3 


267-0 


283-7 


300-4 


H 


182-9 


201-2 


219-5 


237-8 


256-1 


274-4 


292-7 


311-0 


329-3 i 


4 


200-8 


220-8 


240-9 


261-2 


281-1 


301-1 


321-2 


341-3 


361-4' 


H 


219-4 


241-4 


263-4 


285-3 


307-2 


329-2 


351-1 


373-0 


896-0 


3 


238-9 


262-8 


286-7 


310-5 


334-4 


358-3 


382-2 


406-1 


4300 


H 


259-3 


285-2 


311-1 


337-0 


3630 


388-9 


414-8 


440-7 


466-7 


H 


280-4 


308-4 


33ii-5 


364-5 


392-6 


420-6 


448-6 


476-7 


504-7 


H 


302-4 


332-6 


362-9 


393-1 


423 4 


453-6 


483 8 


514-1 


544-3 


H 


3251 


357-6 


390-1 


422-7 


455-2 


487-7 


520-2 


562-7 


586-2 


H 


348-9 


383-7 


418-6 


455 5 


488-4 


523-3 


558-2 


6931 


627-9 


3f 


373-3 


410-7 


448-0 


4863 


522-6 


560-0 


597-3 


634-6 


672-0 


H 


398-0 


438-6 


478-3 


518-2 


55S-1 


598-0 


637-8 


677-7 


717-6 



106 



Practical Tables. 









TABLE n. 








ROUND IROX. 




Size. 


1ft. 


2 ft. 


3 ft. 


1 
4 ft. 5 ft. 


6 ft 


7 ft. 


8 ft 


9 ft. 


i 






Inch. 


lbs. 


]b3. 


lbs. 


lbs. lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


" 4 


42-5 


84-9 


127-4 


169-9, 212-3 


254 8 


297-2 


339*7 


382-2 




4i 


45-2 


90-3 


135-5 


180-7 2'25-9i 271-0 


316-2 


361-4 


406-6 




4i 


48-0 


95-9 


143-9 


191-8 239-8 287-7 


■ 335-7 


383-6 


431-6 




4f 


50-8 


101-6 


152-4 


' 203-3 254-1 304-9 


1 355-7 


406-5 


457-3 




i 4i 


53-8 


107-5 


161-3 


' 215-0 268-8 322-6 


i 376-3 


430-1 


483-8 




1 H 


56-8 


113-6 


170-4 


227-2 283 9j 340-7 


' 397-5 


454-3 


511-1 




! 4| 


60-0 


119-8 


179-7 


239-6 299-5' 359-4 


419-3 


479-2 


539-1 




4J 


63-1 


126-2 


189-3 


252-4 315-5 378-6 


441-7 


504-8 


567-8 




5 


66-8 


133-5 


200-3 


267-0 333-8 400*5 


467-3 


534-0 


600-8 




5i 


69-7 


139-5 


209-2 


278-9 


j 348-7 418-4 


488-1 


557-8 


627 6 




H 


'73-2 


146-3 


219-5 


292-7 


365-9' 439-0 


512-2 


585-4 


658-5 




5I 


76-7 


153-4 


230-1 


308-8 


383-5 


460-2 


536-9 


613-6 


690-3 




5i 


80-3 


160-6 


240-9 


321-2 


401-5 


481-8 


562-1 


6424 


722-7 




5f 


84 


168-0 


252-0 


336-0 


420-0 


504-0 


588-0 


672 


756-0 




5f 


87-8 


175-^ 


263-3 


351-1 


438-9 


526 7 


614-4 


702-2 


790-0 




H 


91 6 


183-3 


274-9 


366-5 


458-2 


549-8 


641-4 


733-1 


824-7 




6 


95-6 


191-1 


286-7 


332-2 


477-8 


573-3 


668-9 


764-4 


860-0 




6i 


103-7 


207-4 


311-1 


41 4 -S 


518 5 


622-2 


725-9 


829-6 


933-3 




6i 


1122 


224-3 


336 5 


448 6 


560-8 


673-0 


785-1 


897-3 


1009-4 




6| 


121-0 


241-9 


362-9 


483-8 


604-8 


725-8 


846-7 


967-6 


1088-6 




1 


130-0 


2301 


390-1 


520-2 


650-2 


780-3 


910-3 


1040-4 


1170 4 




n 


189-5 


279-1 


418-6 


558-2 697-7 


837-3 


976-8 


1116-4 


1255-9 


1 


'7i 


149-3 


29S-7 


448 


597-3 


741-6 


896-0 


1045-3 


1194-6 


1344 


f 


7f 


159-5 


318-9 


478-4 


637-8 


797 3 


956-7 


1116-2 


1275-6 


1435-1 




8 


'69-9 


339-7 


509-6 


679-4 


849-3 


1019-1 


1189-0 


1358-8 


1528-7 




8i 


180-7 


361 4 


542-1 


722-8 


903 5 


1084-2 


1264-9 


1445-6 


1626-3 




H 


191-8 


383-6 


595-4 


767-2: 959-0 


1150-8 


1342-6 


1534-5 


1726-3 




81 


203 3 


406 5 


609-8 


813-0,1016 3 


12196 


1422-8 


1626-1 


1829-3 




9 


2150 


4301 


645-1 


860-2 


1075-2 


1290-2 


1505-3 


1720-3 


1935-4 





Practical Tables. 



101 



TABLE IL 

ROUND IRON. 



Size. 



Inch. 
4 

4i 
4-1 

H 

4f 

4J 



6 
5f 



6 

H 

6f 



7 



8i 
8i 



10 ft. 



424-6 
451 -7 
479-5 

508-2 
537-6 
567-9 
599-0 
6S0-9 



11 ft. 



667-5 
697-3 
731-7 
767-0 
803-0 
840-0 
877-8 
916-3 



955-5 
1037-0 
1121-6 
1209-6 



lbs. 

467-1 
496-9 
527 6 
559-0 
591-4 
624-7 
658-9 
694-0 



12 ft. 



1300-5 
1395 4 
1493-3 
1594-6 



1698-6 
1809-0 
1918-1 
2032-6 



9 2150-4 



734-8 
767-0 
804-9 
813-7 
883-S 
924-0 
965-5 
1008 



105M 
1140-7 
1233 8 
1330-6 



lbs. 

509-6 
542-1 
575-4 
609 -S 
645-] 

681-c;^ 

718-v^ 
757-1 



13 ft. 



lbs. 

552-0 

587 

623-4 

660-6 

698-9 

738-2 

778-7 

8-20-2 



801 
836 
878-1 
920-4 
963 6 
1008-( 
1053-3 
1099 



1430-5 
15850 
1642-6 
1754-0 



1808-4 
1987-7 
•2109-9 
2235-9 



2365-4 



1146-6 
1244 4 



14 ft. 



lbs. 

694-5 
632-4 
671 3 
7114 

752-6 
795-0 
838-6 
883-8 



867-8 

906-5 

951-2 

997-1 

1044-0 

1092-0 

1141-1 

1191-2 



1242-2 
1348-2 
1345-9 1458-1 
1451-5 1572-5 



1560-6 
1674-5 
1791-9 
19135 



■2038-3 
•21684 
2301-7 
2439-1 



2580-5 



934 5 

976-2 

1024-4 

io;3-8 

1124 3 

1176-0 
1228-9 
1282-9 



15 ft. 



lbs. 

637-0 
677-6 

719-3 
762-2 
806-4 
851 8 
898-5 
946-4 



16 ft. 



lbs. 

676-4 

722-8 
767-2 
813-0 
860-2 
908 
958-1 
1009 



1001-3 
1046 
1097-6 
1150-5 
1204-6 
1260-0 
1316-6 1404-4 
1374-5 1466-1 



1068-0 

1115 

1170-8 

1227-2 

1284-9 

1344-0 



17 ft. 



lbs. 

721-9 
761-0 

815-2 
863-9 
913-9 
965-4 
1018-3 
10726 



1337-7 
1451 9 
1570-2 
1693-4 



1820-7 
1953-6 

2090-6 



1690-6 
1814-1 
1941-3 
2072-9 2232-4 2391-8 



1432-3 
1555*6 
1682-4 
1814-4 



1950-7 
2093 2 
2239-9 



2208-1 
2349-0 
•2493-5 
26424 



2795-0 



2378-0 
2529-7 
2685-3 
2845-6 



3010-6 



2547-8 
2740-4 
2879-1 
3048-9 



1528-8 
1659-3 
1794-6 
1935-4 



1134-8 
1185-4 
1243-9 
1303-9 
1365-2 
1428-0 
1492-2 
1557-8 



18 ft 



lbs. 

764-4 
813-1 
863-1 
914-7 

967-7 i 
1022-21 
1078*2 
11357 



1624-4 
1763-0 
1906-7 
2056-3 



2088-8 2210-8 2340-9 



2232-7 2372-2 
2389 2 2538-6 
2551-3 -2710-8 



1201-5 
1255-2 
1317-1 
1380-6 
1445-5 
15120 
1580-0 
1649-4 



1719-9 

1866-7 
2018-9 
2177-3 



2717-7 
2891-1 
3068 9 
3252-2 



3225-6 3440-6 3655 7 



2887-6 
3071-8 
3260-7 
3455-4 



2511-8 
2687-9 
2870-2 



3067-4 I 
3252-5 I 
3452-5: 
3658-7 ; 



3870-7 i 



108 



Practical Tables. 





TABLE IL 




U t' N 1) 1 R O X . 


Size. 

i - 


1ft. 


2 ft. 
lbs. 


3 ft. 


4 ft. 5 ft. 


6 ft. 


7 ft. 


8 ft. 


( 
9 ft 


Inch. 


lbs. 


Ib3. lbs. 


lbs. 


lbs. 


.... ._ 
lbs. 1 lbs. . 




H 
H 
9f 


! 227-2 
239-6 
252-4 


1 454-3 

f 479-2 

505-8 


681-5 
718-8 
757-1 


908 -C 

958-^ 

1009-5 


>!1135-8 1362-9 
ill98-0|1437-6 
1261-9 1514-3 

1 


1590-1 
1677-2 
1766-6 


18l7-2'2044-4 
1916-8 2156-41 
2019-0 2291-4 




10 
lOi 
lOJ 
10| 


266-3 

278-9 
292-7 
306-8 


o32'6 

557-81 

585 -4! 
60^-6 


798-9 
83t?-8 
878-] 
920-4 


1065-2 
1115-7 
1170-8 
1227-2 


! 
1331-4 1597-7 
1 1394-6 1673-5 
■1463-4 1756-1 
■l534'0 1840-8 


1864-0 
1952-5 
2048-8 
2147-6 


2130-3 2396-6 
2231-42510-3 
2341-5 2634-2 
'2454-4 2761-2 




11 

Ui 

iif 


321-2 
336-0 
351-1 
366-5 


642-4 963-6 
672'0!l008-0 
702-2il053-3 
733-l!l099-6 


1284-9 
1344-0 
1404-4 
1466-1 

J 


1606-1 1927-3 
1680-0 2016-0 
1755-52106-6 
1832-7i2l99-2 


2248-5 
2352-0 
2457-7 
2565-8 


2569-7 

2688 
2808-8 
2932-3 


1 
2890-9 j 
3024-0! 
3159-9! 
3298 8' 




1 

12 j 


382-2 


764-4ill46*5 

1 


1 

528-81 

1 


1911-0 2293-2 

i 


2675-5 


3057-7 


i 
3439-9; 



I 



I Bronzing Liquids, F0r Bronzing Copper Medals, Figures, Instru- 
j MENTs, <fec. — 1. Sal ammoniac 1 dr., oxalic acid 15 gr., vinegar 1 
j pint. After well cleaning .the article to be bronzed, warm it gently, 
I and brush it over with the liquid, using only a small quantity at a 
time. When rubbed dry, repeat the application till the desired 
tint is obtained. [For copper medals, electrotype casts, (fee] 

2. Sal ammoniac 1 oz., cream ef tartar 3 oz , salt 6 oz. Dissolve 
in a pint of hot water, add 2 oz. of nitre, and 2 oz. of nitrate of 
copper dissolved in -| pint of water. 

3. Salt of sorrel 1 oz., sal ammoniac 2 oz., white vinegar 14 oz. 
[To give an antique appearance to bronze figures, &c.] 

4. A diluted solution of muriate of platina. [For copper binding 
screws, and other small articles.] 

5. A Vv^eak solution of kydro-sulphuret of ammonia, or of sul- 
phuret of potassium. [For electrotype medals. Another method 
is the following: Immediately oa removing the electrotype cast 
from the solution, brush it over with good black lead ; then heat it 
moderately, and brush it over with a painting brush, the slightest 
moisture being used.] 



P R A CT ic A L Tab le.<. 



109 



















TABLE II. 


ROUND IRON. 


Size. 


10 ft. 


11 ft. 


12 ft 


13 ft. 


14 ft. 

• 


15 ft. 


16 ft. 


17 ft. 


18 ft. 


j' 

i Inch. 


lbs. 


lbs. 


ibs. 


lb?. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


n 


2-271-5 
2396-0 
2523-8 


2498-7 
26356; 

2776-1 


-2725-8 
2875-2 
3028-5 


2953-0 
3114-8 
3280-9 


3180-1 
3354-4 
3533-3 


3407-3 
3594-0 
3785-6 


3634-4 
3833-6 
4038-0 


3861-6 
4073-2 
4290-4 


4088-7 
4312-8 
4542-8 


10 
lOi 
lOi 
10^ 


2662-9 
2789-2 
2926-9 
:^>068-0 


2929-2 
3068-2 
3219-6 

3374-8 


3195-5 
3347-1 
3512-3 

3681-6 


3461-7 
3626-U 
3804-9 
3988-4 


37 '28 
3904-9 
4097-6 
4295-2 


3994-3 
4183-8 
4390-3 
4602-0 


4260-6 

4462-8 
4683-0 
4908-8 


4526-9 4793-2 
4741-7 5020-6 

4975-7o268-4 
5215-6 5522-4 


11 

lli 

iii 


3212-2 
3360-0 
3511-0 

3665-4 


3533-4 
36960 
3862-1 
4031-9 


3854-6 
4032-0 
4213-2 

4398-4 


4175-8 
4368-1 
4564-4 

4765-0 


4497-0 
4704-1 
4915-5 
5131-5 


4818-2 
5040-1 
5266-6 
5498-0 


5139-5 
5376-1 
5619-7 

5864-6 


5460-7 5781-9 
5712-1 6048-1 
5968-8 6319-9 
6231-1 6597-6 


.■•1:2 


3822-1 


4204-3 


4586-5 


4968-7 


5350 9 


5733-1 


6115-3 


6497-5:6879-7 

i 
! 



Solutions used in Elegtrotvpe Manipulations, &c. 

1. Acid Solution of Copper for the Decomposing Cell. Saturated 
solution of sulphate of copper 2 parts, sulphuric acid 2 parts, water 
6 or 8 parts. 

2. Gold Solution. Dissolve 2 oz. of cyanide of potassium (by 
Liebig's method) in a pint of warm distilled water, add ^ oz. of 
oxide of gold, and agitate together. 

3. Silver Solutio7i. Dissolve 2 oz. of Liebig's cyanide of potas- 
sium in a pint of distilled water ; add i oz. of moist oxide of silver 
(precipitated by lime water from a solution of the crystallized 
nitre), and agitate together till the oxide is dissolved. 

4. Solution in which Steel Articles are dipped before Electroplating 
them. Nitrate of silver 1 part, nitrate of mercury 1 part^ nitric 
acid (sp. gr. 1 -384) 4 parts, water 1 20 parts. 

5. Solution, or Pickle, for immersing Copper Articles in before 
Electroplating. Sulphuric acid 61 parts, w;iter 64, nitric acid 32, 
muriatic acid 1. Mix. The article, free from grease, is dip[>ed in 
the pickle for a second or two. 



10 



110 






Practical Tables. 
















TABLE III. 
















FLAT 


IRON 














1 ft. 


2 ft. 


3 ft. 


4 ft. 


5 ft. 


6 ft. 


7 ft. 


8 ft. 


9 ft. 


l 


"^ 




















in. 


in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


i 


1 


0-8 


1-7 


2-5 


3-4 


4-2 


5i 


5-9 


6-8 


7-6 


i 


H 


1-1 


2-1 


3-2 


4-2 


5-3 


6-3 


7-4 


8-4 


9-5 


i 


H 


1-3 


2-5 


3-8 


51 


6-3 


7-6 


8-9 


10-1 


11-4 


i 


If 


1-5 


3 


4-4 


5-9 


7-4 


8-9 


10-4 


11-8 


133 


i 


2 


I'l 


3 4 


51 


6-8 


8-5 


10-] 


11-8 


13-5 


15-2 


i 


2i 


1-9 


3-8 


6-7 


7-6 


9-5 


11-4 


13-3 


15-2 


17-1 


i 


2i 


21 


4-2 


6-3 


8-4 


10-6 


12-7 


14-8 


169 


190| 


i 


2f 


2-3 


4-6 


7-0 


9-3 


11-6 


13-9 


16-3 


18-6 


20-9 


i 


3 


2-5 


5-1 


7-6 


10*1 


12'7 


15-2 


17-7 


20-3 


22-8 


i 


3i 


2-7 


5-5 


8-2 


11-0 


13-7 


16 6 


19-2 


220 


24-7 


i 


^i 


3-0 


6-9 


8-9 


11-8 


14-8 


17-7 


20-7 


23-7 


26-6 


i 


3f 


3-2 


6-3 


9-5 


12-7 


15-8 


19-0 


22-2 


25-4 


28-5 


i 
i 


4 


3-4 


6 8 


101 


13-5 


16-9 


20-3 


23-7 


27-0 


30-4 


41 


3-6 


7-2 


10-8 


14-4 


180 


21-5 


25-1 


28-7 


323 


■^4 

4i 


3-8 


7-6 


11-4 


15-2 


19-0 


22-8 


266 


30-4 


34 2 


i 


4f 


4-0 


8-0 


12-0 


16-1 


20-1 


24 1 


28-1 


32-1 


36-1 


i 


5 


4-2 


8-4 


127 


16-9 


2M 


25-8 


29-6 


33-8 


38-0 


i 


5i 


4-4 


8-9 


13-3 


17*7 


22-2 


26-6 


311 


35-5 


39-9 


i 


oi 


4-6 


9-3 


13-9 


186 


23-2 


27-9 


32-5 


37-2 


41-8 


i 


5f 


4-9 


9 7 


14-6 


19-4 


24-3 


29'2 


34-0 


38-9 


43 '7 


i 


6 


5-1 


10-1 


15-2 


20-3 


25-3 


30-4 


355 


40-6 


45-6 


i 


1 


1-3 


2-5 


3-8 


5-1 


6-3 


7-6 


8-9 


10-1 


11-4 


f 


u 


1-6 


3 2 


4-8 


6-3 


7-9 


9-5 


11-1 


12-7 


14-3 


1 


H 


1-9 


3-8 


5-7 


7-6 


9-5 


11-4 


13-3 


15 2 


171 


f 


If 


2-2 


4-4 


6-7 


8-9 


IM 


13-3 


15-5 


17 7 


20-0 


f 


2 


2-5 


5-1 


7-6 


10-1 


12-7 


15-2 


17-7 


20-3 


22-8 


■1 


2i 


2-9 


5-7 


8-3 


11-4 


14-3 


17-1 


20 


22-8 


25 7 


f 


2i 


3-2 


6-8 


9-5 


12-7 


15-8 


19-0 


22-2 


25-4 


28-5 

! 



Practical Tables. 



Ill 



TABLE III. 

FLAT IRON. 



44 

'r] 


^ 
-»-» 
t: 


10 ft. 


11 ft. 


12 ft. 


13 ft. 


14 ft. 


15 ft. 


16 ft 


17 ft 


18 ft 


H 


^ 




















in. 


in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


i 


1 


8-5 


9-3 


10-1 


11-0 


11-8 


12 7 


135 


14-4 


15-2 


i 


li 


10-6 


11-6 


12-7 


13-7 


14-8 


15-8 


169 


17-9 


190 


i 


n 


12-7 


13-9 


15-2 


16-5 


17-7 


19-0 


20-3 


21-5 


22-8 


i 


If 


U'S 


16-3 


17-7 


19-2 


20-7 


22-2 


23-7 


25-1 


26-6 


i 


2 


16-9 


18-6 


20-3 


22-0 


23-7 


25-4 


27-0 


28-7 


30-4 


i 


2i 


190 


20-9 


22-8 


24-7 


26-6 


28-5 


30-4 


82-3 


34-2 


i 


2i, 


21 1 


23-2 


25-3 


27-5 


29-6 


81-7 


83-8 


35-9 


38-0 


i 


2| 


23-2 


25-6 


27 9 


30-2 


32-5 


84-9 


87-2 


39-5 


41-8 


i 


3 


25-3 


27-9 


30-4 


33-0 


85-5 


38-0 


40-6 


43-1 


45-6 


i 


Si 


27-5 


30-2 


33-0 


35-7 


88-5 


41-3 


43-9 


46-7 


49-4 


i 


H 


29-6 


32-5 


35-5 


38-5 


41-4 


44-4 


47-3 


50-3 


53-2 


i 


H 


317 


34-9 


38-0 


41-2 


44-4 


47-5 


50-7 


53-9 


57-0 


i 


4 


33-8 


37-2 


40-6 


43-9 


47 3 


60-7 


54-1 


57-5 


60-8 


i 


H 


35-9 


39-5 


43-1 


46-7 


50-3 


53-9 


57-5 


61-0 


64-6 


i 


H 


38-0 


41-8 


45-6 


49-4 


53-2 


57 


60-8 


64-6 


68-4 


i 


4f 


401 


44-1 


48-2 


52-2 


56-2 


60-2 


64-2 


68-2 


72-2 


i 


5 


422 


46-5 


50-7 


54-9 


59-1 


63-4 


65-6 


71-8 


76-0 


i 


5i 


44-4 


48-8 


532 


57-7 


62-1 


66-5 


71-0 


75-4 


79-9 


i 


5i 


46-5 


51 1 


55-8 


60-4 


65-1 


69-7 


74 4 


790 


83-6 


i 


5f 


48-6 


53-4 


58-3 


63-2 


68-0 


729 


77-7 


82-6 


87-5 


i 


6 


60-7 


55 8 


60-8 


65-9 


70-9 


76-0 


81-1 


86-2 


91-2 


i 


1 


12-7 


13-9 


15-2 


16-5 


177 


19-0 


20-3 


21-5 


22-8 


t 


u 


15-8 


17-4 


190 


206 


22 2 


23-8 


25-3 


289 


28-5 


t 


H 


190 


20-9 


22-8 


24-7 


26-6 


285 


30-4 


32-3 


34-2 


i 


If 


22 2 


24-4 


26-6 


28-8 


811 


33-3 


35-5 


37-7 


39-9 


i 


2 


25-3 


279 


30-4 


83 


85-6 


380 


40-6 


43-1 


456 


1- 


2i 


28-5 


314 


34 2 


87-1 


39-9 


42 8 


46-6 


48-5 


51-3 


f 


2i 


31 -7! 

1 


34-9 


38 

1 


41-2 


444 


47-5 


50-7 


53-9 


57 



112 



RACTicAL Tables. 



TABLE III. 



FLAT IRON. 



1 ft. 



3-5 

3-8 
4-1 
4-4 

4-8 

5 1 

5-4 
o'l 
6-0 

6-3 
6-7 
I'O 
I'd 

1-6 

1-7 
2-1 



H 


2-5 


If 


3-0 


2 


3-4 


H 


3-8 


H 


4-2 


H 


4-6 


3 


5-1 


H 


5-5 


H 


5-9 


H 


63 


4 


6-8 



2 ft. 

lbs. 

7-0 

7-6 
8-2 
8-9 



10-1 
10-8 
11-4 
12-0 

12-7 
133 
13-9 
14-6 

15-2 

3-4 
4-2 
5-1 
5-9 



7-6 
8-4 
9-3 

10-1 
11-0 
11-8 
12-7 

13-5 



3 ft. 



lbs 
10-5 

11-4 
12-4 
13-3 
14-3 

15-2 
16-1 



17' 
18' 



19-0 
20-0 
20 
21-9 

22-8 

5-1 

6-3 
7-6 
8-9 

10-1 
11-4 

12-7 
13-9 

15-2 
]6'5 
17-7 
19-0 

20-3 



4 ft. 



lbs. 
13-9 

15-2 
16-5 

17-7 
19-0 

20-3 
21-0 
22-8 
24-1 

25-3 
26-6 
27-9 
29 2 

30-4 

6-1 

8-4 

10-1 

11-8 

13-5 
15-2 
16-9 
18-6 

20-3 
22 
23-7 
25-3 

27-0 



5 ft. 



lbs. 

17-4 

19-0 
20-6 
22-2 
23-8 



ft. 



lbs. 
20-9 

22-8 
24-7 
26-6 
28-5 



25-8 


30-4 


26-9 


32-3 


28-5 


34-2 


30-1 


36 1 


31-7 


38-0 


33-3 


39-9 


34-9 


41-8 


86-4 


43-7 


38-0 


45-6 


8-5 


10-1 


10-6 


12-7 


12-7 


15-2 


14-S 


17-7 


16-9 


20-3 


19 ( 


22-8 


21i 


25-3 


23-2 


27-9 


25-S 


30-4 


27-5 


32-9 


29 6 


35-5 


31-7 


880 


33-8 


40-6 



7 ft. 

lbs. 
24-4 

26-6 
28-8 
311 
33-3 

35-5 
37-7 
39-9 
42-1 

44-4 
46-6 

48-8 
51-0 

53*2 

11-8 
14-8 
]7-7 

20-7 

23-7 
26-6 
29 
32-5 

35-5 
38-4 
41-4 
44-4 

47-3 



8 ft. 

lbs. 

27-^ 

30-4 
33 
35-5 

38-0 

40-6 
43-1 
45-6 
48-2 

50-7 
53-2 
55-8 
58-3 

60-8 

13-5 15-2I 

16-9 19-0| 

20-3| 22-8 j 

23 -71 26-6 



27 
30-4 
33 
37 2 



54-1 



40-6 


45-6 




43-9 


49-4 




47-3 


53-2 




60-7 


57-0 





Practical Tables. 



IVd 













TABLE Hi 


















FLAT 


I R O 


N. 




n3 


iOft. 


11 ft. 


12 ft. 


13 ft. 


14 ft. 


15 ft. 


16 ft. 


17 ft 


18 ft. 




^ 


















lbs. 


in 


in. 


lbs. 


lbs. 


lbs. 


lbs. 


lb.?. 


lbs. 


lbs. 


lbs. 


1 


2* 


34-c 


36-g 


41-8 


45-c 


48-^ 


\ 52-3 


55-^ 


i 69-^ 


\ 62-7 


1 


3 


38 -C 


41-8 


45-6 


49-4 


53-^ 


57-0 


60-^ 


64-e 


) 68-4 


i 


H 


41-2 


45-8 


49 4 


53-6 


57-7 


61-8 


65-c 


70-C 


74-2 


f 


H 


44-4 


48-8 


53-2 


57-7 


62i 


66-5 


71-t 


75-4 


79-9 


f 


3i 


47-5 


52-8 


57-0 


61-8 


66-5 


71-3 


76-C 


80 8 


85-5 


f 


4 


50-7 


55-8 


60-8 


65-9 


70-9 


76-0 


81-1 


86-2 


91-2 


f 


H 


53-9 


59 -o 


64-7 


70-0 


75 4 


80-8 


86-2 


91-6 


97-0 


f 


u 


57-0 


62-'i 


68-4 


74-2 


79-9 


85-6 


91-3 


97-0 


102-7 


f 


4f 


60-2 


66-2 


72-2 


78-3 


84-3 


90-3 


96-3 


102-3 


108-4 


1 





63-3 


69-7 


76-0 


82-4 


88-7 


95*0 


101-4 


107-7 


114-0 


f 


5i 


666 


73-2 


79-8 


86-5 


93-1 


99 8 


106-6 


113-1 


119-8 


1 


5* 


69-7 


76-7 


83-7 


90-6 


97-6 


104-5 


111-6 


118-5 


125-5 


f 


54 


72-1^ 


80-2 


87-5 


94-7 


102-0 


109-3 


116-6 


123-9 


131-2 


i 


tj 


76-0 


83-6 


91-2 


98-9 


106 5 


114-1 


121-7 


129-3 


136-9 


i 


I 


16-t 


18-6 


20-3 


22-0 


23-7 


25-4 


27-0 


28-7 


30-4 


i 


li 


21-J 


23-2 


25-3 


27 5 


29-6 


31-7 


33-8 


35-9 


38-0 


i 


H 


25-8 


27-9 


30-4 


33-0 


35-5 


38-0 


406 


43-1 


45-6 i 


\i 


n 


296 


32-5 


35-5 


38-0 


41-4 


44-4 


47-3 


50-3 


53-2 i 


i 


2 


33-8 


37-2 


40-6 


43-9 


47-3 


50-7 


541 


57-5 


60-8 i 


-\ 


2i 


38 -C 


41-8 


45-6 


49-4 


53-2 


57-0 


60-8 


64-6 


68-4i 


i 


'^i 


42-2 


46-5 


50-7 


54 9 


59-1 


63-4 


65-6 


71-8 


76-()i 


i 


n 


46-5 


51 1 


55 8 


60-4 


651 


69-7 


74-4 


79-0 


83 6 1 


i 


8 


50-7 


55-8 


60-8 


65-9 


70-9 


76 


81-1 


86-2 


91-2 


i 


Si 


54-t 


60-4 


65-9 


71-4 


76-9 


82-4 


87-9 


93-3 


98-8! 


i 


3i 


59-2 


65-1 


71-0 


76-9 


828 


88-7 


94-6 


100-6 


106-51 


i 


3f 


63-3 


69-7 


76 


82-4 


88-7 


95-0 


101-4 


107-7 


114-0 


i 


4 


67 -e 


74-4 


841 


87-9 


94-6 


101-4 


108-2 


114-9 


121-7 



10* 



114 



Practical Tables. 









TABLE in. 














FLAT 


I R 


N. 








1 ft. 


'2 ft. 


3 ft. 


4 ft. 


5 ft. 


6 ft. 


7 ft. 


8 ft. 


9 ft. 


H ^ 


















'' 1 


in. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


i H 


7- 


> 14- 


4 21-f 


) 28-'J 


•I 35-{ 


} 431 


50-3 


574 


64-6 


i H 


Y-^ 


> 1/) 


2, 22-^ 


304 


38-( 


) 45-6 


53-2 


60-6 


\ 68-4 


i ^ 


8-C 


) 16 


Ij 24-1 


321 


40-] 


48-2 


56-2 


64 2 


72-2 


i 5 


8-4 


161 


)! 25-3 


33 8 


42i 


> 50-7 


591 


67-6 


760 


' i H 


8-9 


17-'; 


! 26-6 


35-0 


44-^ 


t 53-2 


621 


71-C 


79-9 


\i H 


9-3 


18-f 


) 27-9 


37-2 


46 •£ 


55-8 


651 


74-4 


83-7 


|i 5| 


9-7 


19-4 


\ 29-2 


38-9 


48-6 


•1 58-3 

1 


68-0 


77-7 


87 5 


i « 


10-1 


20 -S 


30-4 


40-6 


! 50-7 


1 60-8 


70-9 


811 


91-2 


f 1 . 


2-1 


4-2 


6-3 


8-4 


10-6 


' 12-7 


14-8 


16-9 


19-0 


f U 


2-6 


0-3 


7-9 


10-6 


13-2 


■ 15-8 


18'5i 211 


23-8 


t li 


8-2 


6-3 


9-0 


12-7 


158 


19-0 


22*2 


25-4 


28-5 


f If 


3 '7 


7-4 


111 


14-8 


18-5 


22'2 


25-9 


29-6 


333 


^ 9 


4-2 


8-4 


127 


. 16-9 


211 


25-3 


29-9 


33-8 


38-0 


1 h 


4-8 


9-5 


14-3 


19-0 


23-8 


28-0 


33-3! 38-0 


42-8 


i 21 


5-3 


10-6 


15-8 


211 


261 


31-7 


37-0 


42-2 


47-5 


t 2f 


5-8 


11-6 


17-4 


23-2 


29-0 


34-8 


40-7 


46-5 


52-3 


i 3 


6-3 


12-7 


19-0 


25-3 


31-7 


38-0 


44-4 


50-7 


57-6 


1 31 


69 


13-7 


20-6 


27-5 


34-3 


41-2 


481 


54-9 


61-8 


1 3| 


7-4 


]4-8 


22-2 


29-6 


37-0 


44-4 


51-8 


59-2 


66-5 


1 3| 


7 9 


158 


23-8 


31-7 


39-6 


47 '5j 


55-5 


63-4 


71-3 


1 4 


8-4 


169 


25-3 


33-8 


42*2 


50-7 


591 


67-6 


76-0 


1 4J 


9-0 


18-0 


26-9' 


3o-9 


449 


53-9 


62-9 71-8 


80-8 


1- 4i 


9-5 


19-0 


28-5: 


38-0 


47-5 


57-0 


66-5 761 


85-6 


I4| 


10 


20-1 


301 


401 


50-2 


60 2 


70-2 


80-3 


90-3 


f 5 


10-6 


2M 


31-7 


42-3 


52-8 


63-4 


73-9 


84-5 


951 


f 5j 


IM 


22-2 


33 -31 


44-4 


55-5 


66-5 


77-6 


88-7 


99 8; 


f 5i 


11-6 


23 2 


34 9; 


46-5 


581 


69-7 


81-3 


92-9 


104-6, 


f 5| 


121 


24-3 


36-4| 


48 6 


60-7 


72-9 


85-0 


97-2 


109-3 











Practical Tables. 






115 










TABLE III. 
















FLAT 


IRON. 








1 


.1 


:5- 


10 ft. 


11 ft. 


12 ft. 


13 ft. 


14 ft. 


15 ft. 


16 ft. 


17 ft. 


18 ft. 


in. 


lbs. 


lbs. 


lbs. 


lbs. 


ibs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


i 


4i 


71-8 


79-0 


86-2 


93-4 


100-5 


107-7 


114-9 


122-1 


129-3 


i 


4^ 


76 


83-6 


91-2 


989 


106-5 


114-1 


121-7 


129-3 


136-9 


i 


4f 


80-3 


88-3 


96-3 


104-3 


112-4 


120-4 


128-4 


136-4 


144-5 


i 


5 


84 5 


92-9 


101-4 


109-8 


118-3 


126-7 


135-2 


143 -C 


152-1 


i 


H 


88-7 


97 6 


106-5 


115-4 


124-2 


133-1 


142-0 


150-8 


159-7 


i 


H 


93-0 


102-2 


111-5 


120-8 


130-1 


139 4 


148-7 


158 


167 3 


i 


5f 


97-2 


106-9 


116-6 


126-3 


1360 


145-8 


155-5 


165-2 


174-9 


i 


6 


101-4 


111-5 


121-7 


131-8 


141-9 


152-1 


162-2 


172-4 


182-5 


i 


1 


2M 


23-2 


25-3 


27-5 


29-6 


31-7 


33-8 


35-9 


38-0 


f 


n 


26-4 


29-0 


31-7 


34-3 


37-0 


39-6 


42-2 


44-^i 


47-5 


1 


H 


31-7 


34-8 


38-0 


41-2 


44-4 


47-5 


50-7 


53-ii 


57-0 


f 


If 


37-0 


40-7 


44-4 


48-1 


51-8 


55-5 


69-2 


62-8 


66-5 


f 


2 


42-2 


46-5 


50-7 


54-9 


60-1 


63-4 


67-6 


71-8 


76-0 


t 


H 


47-5 


52-3 


57-0 


61-8 


66-5 


71-3 


76-0 


80-8 


85-5 


1 


n 


52-8 


58-1 


63-4 


68-6 


73-9 


79-2 


84-5 


89-^ 


95-0 


5 


2| 


581 


63-9 


69-7 


75-5 


81-3 


87-1 


92 9 


98-^ 


104-5 


f 


3 


63-3 


69-7 


76-0 


82-4 


88-7 


95-0 


101-4 


107-7 


114-0 


f 


3^ 


68-7 


75-5 


82-4 


89-3 


96-1 


103-0 


109-9 


116-7 


1236 


1 


3^ 


73-9 


81-3 


88-7 


96-1 


103-5 


110-9 


118-3 


125 7 


133-1 


f 


H 


79-2 


871 


95-1 


103-0 


1109 


1188 


126-8 


134-7 


142-6 


f 


4 


84-/) 


92-9 


101-4 


109-8 


118-3 


126-7 


135-2 


143-('. 


152-1 


t 


4i 


89-8 


98-8 


107-8 


116-7 


125-7 


1347 


143-7 


152-6 


161-6 


1 


4i 


95 1 


104-6 


1141 


1236 


133-1 


1426 


152-1 


161-6 


171-li 


1 


4| 


100-3 


110-4 


120-4 


130-4 


140-5 


150-5 


160-5 


170-6 


180-6 


-1 


5 


105-6 


116-2 


126-8 


137-3 


147-9 


158-4 


169 


179-6 


190-1 


i 


•H 


110-9 


122-0 


133-1 


1442 


155-3 


166-4 


177-5 


188-f> 


199-6 


■1 


♦H 


116-2 


127-8 


139-4 


151-0 


162 6 


174-3 


185-9 


197-5 


209-1 


4 


51 


121-5 
1 


133-6 


145-7 


157 9 


170-0 


182-2 


194-3 


206-5 


218-6 



116 






Practic. 


iL Tables. 
















TABLE III. 
















F 


L A T I R O > 












C3 
n3 


1 ft. 


2 ft. 


3 ft. 


4 ft. 


5 ft. 


6 ft. 


7 ft. 


8 ft. 


9 ft. 


1^ 


^ 




















in. 


in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


f 


6 


12-7 


25-3 


38-0 


50-7 


63-4 


76-0 


88-7 


101-4 


114-1 


f 


1 


2-5 


5-J 


7-6 


10-1 


12-7 


15-2 


17-7 


20-3 


22-8 


f 


H 


3-2 


6-3 


9-5 


12-7 


15-8 


19-0 


22 2 


25-4 


28-5 


f 


H 


3-8 


7-6 


11-4 


15-2 


19-0 


22-8 


26-6 


30-4 


24-2 


f 


i| 


4-4 


8-1 


13-3 


17-7 


22-2 


26-6 


31-1 


35-5 


39-9 


f 


2 


5-1 


]0-l 


15-2 


20-3 


25 3 


30-4 


35-5 


40-6 


45 6 


f 


2i 


5-7 


11-4 


17-1 


22-8 


28-5 


34-2 


39-9 


45-6 


51-3 


f 


H 


6-3 


12-^< 


19-0 


25-3 


31-7 


38-0 


44-4 


50-7 


57-0 


f 


n 


7-0 


131 


20-9 


27-9 


34-9 


41-8 


48-8 


55-8 


62-7 


f 


3 


'7-6 


15-2 


22-8 


30-4 


38-0 


45-6 


53-2 


60-9 


68-4 


f 


3i 


8-2 


16-5 


24-7 


33-0 


41-2 


49-4 


57-7 


65-9 


74-2 


1 


3i 


8-9 


17-7 


26-6 


35-5 


44-4 


53-2 


62-1 


71-0 


79-9 


f 


3f 


9-5 


19-0 


28 5 


38-0 


47-5 


57-0 


66-5 


76-1 


85-6 


f 


4 


10-] 


20-8 


30-4 


40-6 


50-7 


60-8 


70-9 


81-1 


91-2 


f 


4i 


10-8 


21-5 


32-3 


43-1 


53-9 


64-6 


75-4 


86-2 


97-0 


f 


4i 


11-4 


22-8 


34-2 


45-6 


57-0 


68-4 


79-9 


91-3 


102-7 


f 


4f 


12 


24-1 


36-1 


48-2 


60-2 


72-2 


84-3 


96-3 


108-4 


f 


5 


12-7 


25-3 


38-0 


50-7 


63-4 


76-0 


88-7 


101-4 


114-0 


f 


5i 


13 3 


26-6 


39-9 


53-2 


66-5 


79-8 


93 1 


106-5 


119-8 


f 


5i 


13-9 


27-9 


41-8 


55-8 


69-7 


83-7 


97-6 


111-5 


125-5 


f 


5f 


14-6 


29-1 


43-7 


58-3 


72 9 


87-4 


1020 


116-6 


131-2 


f 


6 


15-2 


30-4 


45-6 


60-8 


76-0 


91-2 


lOG-5 


121-7 


136-9 




l-?r 


5-1 


lO-l 


15-2 


20-3 


25-3 


30-4 


35-5 


40-6 


45-6 




2 


6-8 


13-5 


20-3 


27-0 


33-8 


40-6 


47-8 


54-1 


60-8 




3 


10-1 


20-3 


30-4 


40-6 


50-7 


60-8 


70-9 


81-1 


91-2 




4 


135 


27-0 


40-6 


54-1 


67-6 


81-1 


94-6 


108-1 


121-7 




5 


16-9 


33-8 


50-7 


67-6 


84-5 


101-4 


118-3 


135-2 


152-1 




6 


20-3 


40-6 


60-8 


81-1 


101-4 


121-7 


141-9 


162-2 


182-5 













P 


RACTICAL TaDLES. 






117 






TABLE III. 






i 






FLAT 


IRON. 




j 








10 ft. 


11 ft. 


12 ft. 


13 ft. 


14 ft. 


15 ft. 


16 ft. 


1 i 
17 ft. 18 ft. \ 

i 




in. 


in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


i 
lbs. 1 




f 


6 


126-7 


139-4 


152-1 


164-8 


177-4 


190-] 


202-8 


215-4 


228 1 : 

! 




f 


1 


25-3 


27-9 


30-4 


33-0 


35-5 


38-( 


40-6 


43-1 


45-6 j 




f 


H 


31-7 


34-9 


38-0 


41-2 


44-4 


47 -L 


50-7 


53-9 


57-0 




f 


11 


38-0 


41-8 


45-6 


59-4 


53-2 


57 -( 


60-8 


64-6 


68-4 




f 


U 


44-4 


48-8 


53-2 


57-7 


62-1 


^ 66 -.0 


71-0 


75-4 


79-9 




f 





50-7 


55-8 


60-8 


65-9 


70-9 


76-( 


81-1 


86-2 


91-2 




i 


k 


57-0 


62-7 


68-4 


74-2 


79-9 


85-5 


91 3 


97-0 


102-7, 




f 


H 


63-3 


69-7 


76-0 


82-4 


88-7 


9 5 -(J 


101-4 


107-7 


114-0 




f 


H 


69-7 


76-7 


83-7 


90-6 


97-6 


104-5 


111-5 


118-5 


125-5 




f 


3 


76-0 


83 •€ 


91-2 


98-9 


106-5 


114-1 


121-7 


129-3 


136 9 




f 


3i 


82-4 


90-6 


98-9 


107-1 


115-3 


123-6 


131-8 


140-0 


148-3 




f 


H 


88-7 


97-6 


106-5 


115-4 


124-2 


133-J 


142-0 


150-8 


159-7 




f 


H 


95-1 


104-6 


114-1 


123 6 


133-1 


142-0 


152-1 


161-6 


171-1 




1 


4 


101-4 


111-5 


121-7 


131-3 


141-9 


i52-l 


162-2 


172-4 


182-5 




f 


4i 


107-7 


118-5 


129-3 


140-1 


150-8 


161 •(; 


172-4 


1832 


193-9 




f 


4^ 


114-1 


125-5 


136-9 


148-3 


159-7 


171-1 


182-6 


193-9 


205-3 




f 


4f 


120-4 


132 4 


144-5 


156-5 


168-6 


180 6 


192-6 


204-7 


216-7 




f 


5 


126-7 


139-4 


152-1 


164-8 


177-4 


190-1 


202-8 


215-4 


228-1 




f 


5i 


133-1 


146-4 


159-7 


173-0 


186-3 


199-6 


212-9 


226-2 


239-5 




f 


5.V 


139-4 


153-3 


167-3 


181-2 


195-2 


209-2 


223-1 


237-0 


250-9 




f 


si 


145-7 


160-3 


174-9 


189-5 


204-0 


218-0 


233-2 


247-8 


262-3 




f 


6 


152-1 


167-3 


182-5 


197-7 


212-9 


228-1 


243-3 


258-5 


273-7 






1'^ 


50-7 


65-8 


60-8 


65-9 


70-9 


76-0 


81-1 


86-2 


91-2 






2 


67-6 


74-4 


81-1 


87-9 


94-6 


101-4 


108-1 


114 9 


121-7 






3 


101-4 


111-5 


121-7 


131-7 


141-9 


152-1 


162-2 


172-4 


182-5 






4 


135-2 


148-7 


162-2 


175-7 


189-8 


202-8 


216-7 


229 8 


243-3 






5 


169-(i 


185-9 


202-8 


219-7 


236-6 


253-5 


270-4 


287 3 


304-2 






C) 


202-8 


223 1 


243-3 


263-6 


283-9 


304-2 


324-4 


844-7 


365-0 



lis 






Table 


OF Gradients. 










T^ 




)F GRADIENTS 




1 


^BLE ( 




And Resistance per Ihn for 


each. 




Vertical Rise. 


Gravity 


Vertical Rise. 


Gravity 


Vertical Rise. 


Gravity 






due to 
incline 






due to 
incline 






due to 
incline 














Ratio. 


Pr. Mile. 


per ton. 


Ratio. 


Pr.Mile. 


per ton. 


Ratio. 


Pr. Mile. 


per ton. 


one in 


Feet. 


lbs. 


one in 


Feet. 


lbs. 


one in 


Feet. 


lbs. 


100 


52-80 


22-40 


74 


71-38 


32-270 


. 47 


112-34 


47-660 


99 


53-33 


22-626 


73 


72-32 


30-685 


46 


115-04 


48-684 


98 


53-88 


22-858 


72 


73-33 


31-111 


45 


117-33 


49-777 


' 97 


54-48 


23-092 


71 


74-36 


81-550 


44 


120-00 


50-908 


' 96 


55-00 


23-334' 


70 


75-43 


32-000 


43 


122-78 


52-092 


95 


55 60 


23-579| 


69 


76-49 


32-464 


42 


125-71 


53-333 


94 


56-17 


23-830 


C8 


77-64 


32-940 


41 


128-78 


54-634 


93 


56-77 


24-086 


&1 


78-81 


33-432 


40 


182-00 


56-00 


92 


57-52 


24-342 


66 


80-0 


83-940 


39 


185-88 


57-436 


91 


58-02 


24-614 


65 


81-23 


34-460 


88 


188 95 


58-944 


90 


58-66 


24-888 


64 


82-50 


35-0 


87 


142-70 


60-540 


89 


59-33 


25-168 


63 


8:i-81 


35-555 


86 


146-66 


62-222 


88 


60-0 


25-454 


62 


85-16 


36-108 


85 


150-84 


64-000 


87 


60-69 


25-746i 


61 


86-55 


36-720 


34 


155-30 


65-880 


86 


61-39 


26-046! 


60 


88 00 


37-833 


83 


1600 


67-880 


85-16 


62-00 


26-303; 


59 


89-49 


37-966 


32 


165 


70 


85 


62-12 


26-353 


58 


91-03 


88-620 


81 


170-82 


72-216 


84 


62-86 


26-666 


57 


92-63 


39-298 


80 


176-00 


74-666 


83 


63-61 


26-988 


.56. 


94-28 


40-0 


29 


182-06 


77-240 


82 


64-39 


27-317 


55 


96-00 


40-726 


28 


188-56 


80-00 


81 


65-20 


27-718 


54 


97-77 


41-480 


27 


195-55 


82-960 


80 


66-0 


28-00 


53 


99 62 


42-284 


26 


208-06 


86-152 


79 


66-83 


28-355 


52 


101-58 


48-076 


25 


211-20 


89-60 


78 


67-69 


28-718 


51 


103-52 


43-920 


24 


220-0 


93-336 


77 


68-57 


29-090 


50 


105-60 


44 800 


23 


229-56 


97-368 


76 


69-47 


29-472 


49 


107-75 


45-716 


22 


240- 


101-816 


75 


70-40 


29-867 


48 


110-00 


46-688 


21 


251-43 


106-666 


To T 


\KE Impression 


S FROM 


Coins, &c. — M 


ake a tl 


lick solution of 


1 isinglas 


s in wj 


iter, an 


i lay it 


hot o 


[1 the m 


etal; h 


it it rei 


nain for 



twelve hours, then remove it, breathe on it, and apply gold or 
silver-leaf on the wrong side. Any color may be given to the isin- 
glass instead of gold or silver, by simple mixture. 

Variations in Tides. — The difference in time between high water 
averages about 49 minutes each day. 







Cast- 


Iron Pillarj^. 




119 


TABLE 


f the Ultimate Break 


:ing Weight, in tons, 


of cast-iron pillars, 


calculated from Professor Hodgkhison s 


Formula. 




The length includes every half-foot from 1 to 20, and the diameter every inch from 1 to 24. 


H tJ 


DIAMETER OF CAST-IRON PILLAR 


S IN INCHES. 


O >?; w 














•K, '-^ ^ 














t-1 


1 


2 


3 


4 


5 


6 




tons. 


tons. 


tons. 


tons. 


tons. 


tons- 


1 


44-30 


537 


2312 


6513 


14544 


28038 


H 


22-23 


269 


1160 


3269 


7300 


14073 


2 


13-63 


165 


711 


2004 


4476 


8630 


2i 


9-33 


113 


487 


1372 


3064 


5905 


3 


6-84 


83 


357 


1006 


2247 


4331 


H 


5-26 


64 


275 


774 


1729 


3333 


4 


4-19 


51 


219 


617 


1378 


2656 


4i 


3-43 


41-6 


179 


505 


1127 


2174 


5 


2-87 


34-8 


150 


422 


943 


1817 


H 


2-44 


29^6 


127 


359 


802 


1545 


6 


2-11 


25^5 


110 


309 


691 


1333 


61 


1-84 


22^3 


96 


270^ 


603 


1163 


7 


1-62 


19-6 


84-6 


238 


532 


1026 


7i 


1-44 


17^5 


75-2 


212 


473 


912 


8 


1-29 


15-6 


67-4 


190 


424 


817 


81 


1-16 


14-] 


60-8 


171 


382 


737 


9 


1-06 


12-8 


55-2 


155 


847 


669 


H 


-96 


11-7 


50^3 


142 


316 


610 


10 


•88 


10-7 


461 


130 


290 


559 


101 


•81 


9-86 


42^4 


119 


267 


515 


11 


-75 


9^11 


39-2 


110 


24G 


475 


llj 


•69 


8^45 


36^3 


102 


228 


441 


12 


-65 


7^86 


33-8 


95-3 


212 


410 


121 


•60 


7-33 


31^5 


88-9 


198 


383 


13 


•56 


6-86 


29-5 


83-2 


185 


358 


131 


•53 


6^43 


27-7 


78^0 


174 


336 


U 


•50 


6^05 


26^0 


73-3 


163 


315 


141 


•47 


5-70 


24^5 


69^1 


154 


297 


15 


•44 


5^38 


23^15 


65-23 


145^6 


280-8 


151 


•42 


5^09 


21-90 


61-69 


137^7 


265-5 


16 


-40 


4^82 


20-75 


58-45 


130-5 


251-6 


161 


•377 


4^57 


19-69 


65^47 


123-8 


238-8 


\n 


•358 


4-35 


18^72 


52-73 


117-7 


2270 


m 


•341 


4^14 


17^82 


50-19 


112-1 


216-1 


18 


•325 


3-94 


16^98 


47-85 


106-8 


205-9 


181 


•310 


3^77 


16^21 


45-67 


101-9 


190-6 


19 


•297 


3^60 


15-49 


43-64 


97-45 


187-8 


191 


'284 


3-44 


14-82 


4176 


93^24 


179-7 


20 


•272 


3-30 


11-20 


40^00 


89-32 


172-2 









120 



Cast-Iiion Pillars. 



TABLE of the Ultimate Breaking We'-ght, in tons, of cast-iron pillariL 
(Continued.) X 



?5 " j^ 


D1A3IETER OF CAST IROX PILLARS IN INXHES. 












I 




7 


8 


9 


10 

tons. 


1 ^^ 


12, 




Ions. 


tons. 


tons. 


': tons. 


tons. 


1 


48838 


78982 


-120691 


176361 


' 248552 


339985 


u 


24513 


39643 


60579 


88520 


1 124756 


170648 


2 


15031 


24310 


37147 


; 54282 


I 76501 


104643 


2i 


10286 


16635 


25420 


' 37145 


i 52350 


71607 


3 


7544 


3 2202 


18645 


! 27246 


i 38398 


52523 


3i 


5805 


9388 


14347 


20965 


29546 


40414 


4 


4626 


7482 


11433 


16707 


23546 


32207 


•ii 


3787 


6124 


9358 


13675 


19273 


26363 


5 


3166 


5120 


7824 


11433 


16113 


22039 


5^ 


2692 


4354 


6653 


9722 


13703 


1S743 


6 


2322 


3755 


5738 


8385 


11818 


16165 


6i 


2026 


3277 


5008 


7319 


10315 


14109 


7 


1787 


2889 


4415 


6452 


9094 


12439 


n 


1589 


2570 


3927 


57S8 


8087 


11062 


8 


1424 


?302 


3519 


5142 


7247 


9913 


8^ 


1284 


2077 


3174 


4638 


6537 


8942 


9 


1165 


1885 


2880 


4209 


5932 


8114 


n 


1063 


1719 


262^7 


3839 


5411 


7401 


10 


974 


1575 


2408 


3519 


4959 


6783 


10* 


897 


1450 


2216 


3238 


4564 


6243 


11 


828 


1340 


2048 


2992 


4217 


5769 


\\\ 


768 


1242 


1898 


2774 


3910 


5349 


12 


714 


1156 . 


1766 


2581 


3637 


4975 


124- 


666 


1078 


1647 


2408 


3393 


4642 


13 


623 


1008 


1541 


2252 


3174 


4343 


13i 


585 


946 


1445 


2112 


2977 


4073 


14 


550 


889 


1359 


1986 


2799 


3828 


14i 


518 


838 


32S0 


1871 


2637 


3607 


15 


489-1 


791-0 


1208 


1766 


2489 


3405 


15i 


462-6 


'748-1 


1143 


1671 


2354 


3220 


16 


438-3 


708-8 


1083 


1583 


2230 


3001 


161 


415-9 


672-6 


1028 


1502 


2117 


2895 


17 


395-3 


639-4 


977-0 


1428 


2012 


27^ 


in 


37 G -3 


608-6 


930-1 


1359 


1915 


2620 


18 


35S-7 


580-2 


886-5 


1295 ! 


1826 


2497 


18* 


342-4 


553 8 


846-2 


1236 , 


1743 


23^4 


19 


327-2 


529-2 


808-7. 


1182 : 


1665 ! 


2278 


19* 


313-1 


506-4 


773-8, 


1131 , 


1593 1 


2179 


20 


299-9 


485-0 


741-2 


1083 


1526 i 


2088 


Note.— Ex 


amp^e. Find 


the breaking 


weight of a f 


iast-iron pilla 


r whose exte 


mal diame- 


iterislT, and 


internal diaj 


meter 15 inch 


es, and lengt 


h 13 feet. 











CastJron Pillars. 




121 


TABLE of the Ultimate Breahhig Weight, in tons 


of cast-iron pillars. 








(Continued. 








LENGTH 

IN 

FEET. 


DIAMETER OF CAST-IRON PILLARS IN INCHES. 


13 


14 


15 


IG 


17 


18 




tons. 


tons. 


tons. 


tons. 


tons. 


tons. 


I 


453524 


592195 


759158 


957714 


1191290 


1463470 


H 


227638 


297241 


381039 


480707 


597950 


734563 


2 


139588 


182269 


233660 


294769 


366664 


450443 


2i 


95522 


124729 


159895 


201717 


250912 


308238 


3 


70064 


91486 


117281 


147955 


184040 


226088 


H 


53912 


70396 


90243 


113846 


141614 


173966 


4 


42963 


56100 


71917 


90726 


112853 


138638 


4i 


35137 


45920 


58867 


74263 


92375 


113481 


6 


29400 


38390 


49213 


62085 


77228 


94871 


^ 


25002 


32647 


41851 


52798 


65676 


80680 


6 


21565 


28158 


36097 


45538 


56645 


69586 


H 


18821 


24576 


31505 


39745 


49439 


60734 


7 


16593 


21667 


27776 


35040 


43587 


53545 


7^1 


14756 


19269 


24701 


31163 


38763 


47619 


8 


13223 


17267 


22135 


27924 


34735 


42671 


8^ 


11928 


15576 


19967 


25190 


31333 


38492 


9 


10824 


14133 


18118 


22857 


28432 


34928 


n 


9873 


12892 


16527 


20850 


25935 


31861 


10 


9049 


11815 


15147 


19109 


23769 


29200 


10^ 


8329 


10875 


13941 


17588 


21877 


26876 


11 


7695 


10048 


12882 


16250 


20214 


24832 


IH 


7135 


9317 


11944 


15067 


18743 


23025 


12 


6637 


8667 


11110 


14016 


17434 


21418 


12^ 


6192 


8086 


10365 


13076 


16265 


19982 


13 


5793 


7564 


9697 


12233 


15216 


18693 


13^ 


5433 


7094 


9094 


11472 


14271 


17531 


14 


6107 


6669 


8549 


10785 


13415 


16481 


141 


4811 


6282 


8054 


10160 


12638 


15526 


15 


4542 


5931 


7603 


9591 


11930 


14656 


15^ 


4296 


5609 


7191 


9071 


11283 


13862 


16 


4070 


5314 


6813 


8595 


10691 


13103 


16^ 


3863 


5044 


6466 


8157 


10146 


12464 


17 


3671 


4794 


6146 


7753 


9424 


11847 


17i 


3495 


4563 


5850 


7380 


9180 


11277 


18 


3331 


4350 


5577 


7035 


8571 


10750 


18^ 


3180 


4152 


5323 


6715 


8353 


10261 


19 


3039 


3968 


5087 


6417 


7983 


9806 


19.J- 


2908 


3797 


4807 


6140 


7638 


9383 


20 


2785 


3637 


4662 


5881 


7316 


8987 


Alon? the 


line marked 


18 feet, and i 


n the vertica 


1 lines nunrib 


iredlTnnd 15 


inches, tnke 


the number 


8 8751 and 5 


577 ; the diffe 


rence of vvh 


ch, namely 3 


174, will be 


the breaking 



11 



122 



Cast^ron Pillars. 





TABLE o//A<? Ultimate Breaking Weight, in ions, 


of cast-Iron Pillars. 










(Continued.) 








^. "-I U3 


DIAMETER OP CAST IRON PILLARS IN INCHES. 




19 


20 


21 


22 

i tons. 


23 

tons. 


24 

tons. 






tons. 


tons. 


tons. 




1 


1777940 


2138510 


2549140 


O013880 


3536910 


4122530 




0O..44 


892404 


1073380 


1279490 


1512760 


1775280 


2069230 




?;-:2:. . 


547224 


658204 


784589 


927630 


1088610 


1268880 




r,. .2f : 


-374471 


450416 


536902 


634786 


744947 


8^8292 




^0 c8:j ; 


'274670 


330374 


393810 


465605 


546409 


636880 




o^■ W ! 


1311350 


254212 


303024 


358269 


420444 


490059 




8..>;4-f ' 


(16842« 


202586 


241485 


285511 


335059 


390543 


' 


:4^ 


137865 


165825 


197666 


233703 


274260 


319671 




5 


115257 


138632 


165251 


195378 


229286 


267248 




5* 


98017 


117894 


140532 


166157 


194988 


227273 




6 


84539 


101684 


121210 


143307 


168177 


196023 




6^ 


73784 


88748 


105789 


125047 


146781 


171085 




7" 


65051 


78243 


93266 


110270 


129406 


150832 




7i 


57851 


69584 


82944 


98067 


115085 


134140 




8 


51840 


62353 


74326 


87876 


103126 


120200 




8i 


46763 


56247 


67047 


79271 


93028 


108430 




9 


42433 


51038 


60840 


71930 


84414 


98390 




9i 


38707 


46557 


55496 


65614 


77000 


89750 




10 


35474 


42669 


50862 


60134 


70571 


82255 




lOi 


32651 


39272 


46814 


55348 


64954 


75708 




i 11 


30168 


36286 


43254 


: 51140 


60014 


69951 




1 lU 


27973 


33645 


40106 


47417 


55646 


64860 




12 


26020 


31297 


37306 


44108 


51763 


60333 




12i 


24275 


29199 


34805 


41150 


.4&292 


56288 




13 


22710 


27315 


32560 


38497 


45178 


52658 




13i 


21298 


25618 


3">537 


36104 


42370 


49385 




14 


20021 


24082 


28706 


33940 


, 39830 


46424 




14i 


18862 


22687 


27043 


31974 


37523 


43736 




15 


17806 


21417 


25529 


! 30184 


! 35421 


4128& 




: 151 


16840 


20255 


24145 


i 28547 


; 33501 


39049 




16 


15955 


19191 


22823 


! 27047 


i 31740 


36997 




16i 


15142 


18213 


21711 


; 25669 


30123 


31111 ; 




17 


14393 


17312 


20636 


i 24398 


28632 


SS374 




i in 


13701 


1 164S0 


19644 


i 23225 


1 27255 


31768 




18 


13060 


15709 


18725 


22139 


i 25981 


80283 




18* 


12466 


14994 


17873 


21131 


! 24799 


28906 




19 


11913 


14330 


17081 


20195 


j 23700 


27624 




in 


' 11398 


13710 


1 16343 


; 19322 


i 22676 


26430 




20 


10918 


1313^ 


1 15654 


• 18508 


21721 


25317 


■weight in tc 


)ns. For pra 


ctifvil purpos 


65 the pillars should be 


calculated to 


bear one half 




more than l 


he weight to 


which they 


are subjected. 























Cast-Iron Shafts. 



123 





^-^ 








q;) C 








CO a; 








?.§ ^ 








r* ^ ^ 








.S ? 2 














^ ^ 5 








i^^ 








c3 rH 
































^ 1 - 








- •^'^'^ 
















So-"" 








j'^r->^' 






















ryj 


<«^:3 






S 


cs cu :3 






< 










p = CU 




p 


f-i r^ 4-^ 












k~( 


^ 


^-z ^ 




rq 


o 


^ :z "Z 




ci 


t— 1 


13 ^ — 






g 






to 


a; — r/3 






-<I 








d 


-J i 




c^ 


o 


^J 3 ^ 




O 




1^. 1 




O 


H 


CC C 






O 
^ 




S 


o 


W 




o 


ti 




o 


PA 




r- "^ 




o 




k! 


o 


'^ -i^ :z 




W 




? c cj-j 






w 




"—^ 


^ 


tJ 


^ (/J a- 


<v 


P 


•3 




Si 


^< 


&^ 




bf) 






, O JL 
















Q? re ^O 
















<^ 3 C3 


n 






















ct "5 ^ 


o 














^ "o ^ 


cy 






<V -f, -^ 


■-^ 






^ ?^ C3 
















<4^ ^ '^^ 








C ^ So 


ri3 






-sil 


'/I 





^Jz ? 



^Xt^0CH5^tiQnSO|0O<4» Ht* H»q CC)^ r-^Tf COHl r4» 

J-' CO CO '^ rf iQ iO CO «£) O O i— £- QO 






o 



j-: CO TT »^0 ^ O O O O i^ i.-^ l— 00 GO 



05 


d 


COrt<iOOO<:o<:Oi:-X:-X--ODOOC55 






o 


.S 


T-|c^^ r^l^ Hcc .-4» Mjoo kH ccjoo uTjx thI^ kH* Ht* 

cOTtiioio?oo<:oi--j:-i:-oooo05 


CO 


.S 


CO rj^ lO O CO CO CO i:- J:- I:- GO 05' Oi 



i.':|oo in|a) ^tJi t-J» ^tJi «4» os|oo u-^ -i^< r-^oo r^ii 
c c0Tt<i0OC0c0i:-ir-x:-G0QQC^05 



I u:|coH'^ solvit HwrH(?it-|ooH'^--*|y>r-JQoecH!H-* 

j£ ThiiO»oco<:oi.--i>-J:^c»oo05050 



,-(f» H® «|x o:^ OT^jt r-joo i-(|ei ecH' Mjco r-^ 
^10 0COJ:-i.--Ir-OOG0000500 



ccjao ccjoo r-iir;i< t-|ao ccjoo c-|aD iH|tji "oIoo m|tj< ic|qo -^ 

Tti»ocioj.-j:-ooooc5C500— ' 



r-ijoo TlosscjooHco o:|'«J< HtJi ecH* H-* 'clx t-joo lol* r-nl-* 
lOCOl-^COOOO^'OSOO'-i—i CM CO 



ecjx Hoo H« »n|a) H-* H® «*» H=o H"** *=H* "^ =^ H» 
iOCOt^CX)OiCiOO'— ^^^MCO-!^ 



c:,yj ijpo 0-* ««|ao H-* «Hi r-*xi irrjco Hm m|xi r-^ 
O !■- CO O O O i-H r-i CM 'M CO TT lO 






lU 



Cast-Isox Soafts. 



< 



c 



f OD 



I I 



o5 


1 ^- 


x-x^^'^??--r7r^Tf;?n- 




r^ 


: = 


x'ci ^ri*c c'— ^"^TTrTrt rc^^rT 


j !^ 


i _- 

i -= 


acr ^ ^ -^ ^ 3= =>* ^Tt Off ^ =» oc* -»« 


^ 


^ 


-i-»cie^ =sr «s: -cz =M -<x oct ^ =3^ =n> -ir aK »♦ . 


^ -^ j 


^: 


=' 


^-^^^''^=^^^ ffr7z*^^^:^^ ' 




c 


= 


r?C ^c*— ^tT??5C cTr^P'^o 0* 


2: 


= 




•OD 


_• 


■0 - ^— ^'ti rTc?^*— — ~~ T^I?'^ 


^ 


^ S 


^ 


-+^0=0: =w:=>*-*^i^aK^^— =^_:i_s3«*-ct:-Tr* 




^^ 


= 


-■"^IlTfr-T ?f ^r: ^^^ o't^tr""? 






HzrzETiFHzzsS'^i'fi: 


! S 


= 


• g 




=ax-j-K:-*-*atr-.T.^z3tr aa: «-«—** :a^ -er -«» jj 



— ?< M rc c^c — — o ^ ^ t; r- r- X X 



< 



• X- X X X 






r-X XCXCC: — — ?1 







a-*=3Er 


-^-a: 


zscx. 


---a: 


a-r* 


oc 


act 




0(E 


__9 


,c-<* 


sati-x 


-^r> 





- 


-r 


-^"Z 


-^ 


— 


XZ 


x^ 


X 


Ci 


^ 


c 


— 


— ' 


^1 


r. 


rc 








"" 
































^-(r 


_^ 


— tx 


oa: 


—fX 


_CE 




=-* 


— Ci 




e-<Z 


_c- 


— « 


lacE 


_^. 


^ 


_; 


t^ 


^ 


•^ 


■^ 


X 


Ci 


3 




— . 


-71 


~I 


rr 


-* 




3C 


^' 


— 


—" 


*■" 


— 


~" 


— 


~" 


r4 


?4 


?4 


7^ 


^ 


71 


>♦ 


-?! 


?J 



it-<Z -<r« ' w-g -(g — CC — *-:» -—» — <X -;-<K rata ^)X a>* 

■fi t; ti ^* ^1 -ri rj rJ ^i t: r« r^ r? re r? 












Wrougut-Iron Shafts. 



m 



TABLE OF STRENGTHS OF WROUGHT-IRON SHAFTS. 


H 


^ 1 

CO 1 


j; H- Ht* H~ n- L-^ HCC ^1-^ {- W|00 .-!- "^1- -:^ «^ 

— (^^oocort^^^OlOOO«oOI:-l:- 


o 


— (McOTfirt^^^OiOOcococor-i:- 


O 


• -.-.=^;^ -,„ ^Htj, H» ^i- H«^ r^ -l^"|« .«J» -.^ ^1- ccH. 
•-3<)C0rj''*TfiOiOiO»O<:0cOi:-J:-X:- 


o 
o 


C -i- MlrH H^ ^'h- MJ» '4» HSD H« MJaO HOO '"'^1- «^ 

•=cNco'g<Tf»OiO»oioooor:-i:-oo 




-COCOTt^TtiiOiOiOcOOCOir-lr-i— 00 


o 


c Ico-l- cc}» -'-. H^ H?^ Hoo r+o ecloD ^-l- r^ 1- -1- 1- 

•-cocoTr'Tt<ooocC'':ocO£^i>j:'00 


CO 


cj^lM Hot) '^i- ^1- "-^i- -!-. -1^ H-* ^P^ wH H'* -I- 1- -t^i 
— COCOTt<Tj< kCOVOCOOCOlr-Xr-COOO 


o 

QO 


^. "^ico ^1- n- -1- 1- <4» »r4» Hoo n- H* H-* "ol* 
■=COTJHTt<\OiO^O«OCOOX:-i:-0000 




^ ^^ "l„ -i- r*o ^1 -1- ^+* .-4r» kNi -4?' 0*0 J« 
ScOTj^Tj^OlCilOCOCDCCIr-t-OOQOOO 


o 


•=C0-rHT^».0Ut)O«0C0C0r-l>-G0000i 


O 

§ 
1 

O 

cq 
D 


S 


^ H^ "^1 1-.^ ^i„ -'1^ r^r. -1- H« «!* -1- *^kH« -+* 


S 


— cOTti«^iOO?DcOi::-i:-i:-Goa)Gi05 




_.|«5 ^|« _1<0 

^^:„ r^ -;,- «|^ H«3 ^-|- H.J. r^si ecHt 0*0 o*o H» 

— co-^ioxocooj^-r-i^-j^-ooososas 


g 


r; ^4* —1-0*0 0*0 IcH* HoO r-^ cdt^ idfC r-lpo iOCD r-^ 

— C0Ti<oOc0<:0i>-X:-ir-0000a50sO 

I— 1 




^ ^ -^l„ ==|„ ^ =^-|,» Hoo e*o «H( ^1- ojjao Hy» 

— coxt<io«ocoy::x:-x>oocx)asooo 

r-l r-H 


o 


e «,'^ e*o "l- "'f- -« 0*0 M» 1- O "^1- H» 
— M^iOiOcDt-X^Jr-OOQOCOCSi-tOO 


CO 


^|co lOto „|co ^ico cn,co |co 
d 1- H^ -i—sjM "l- w!oo n- -1- ccH* "^ - Hoo 0*0 
— Ti^lO»oo^:-t-oooocx)050:)00-' 


o 

CO 


J£ 0*0 "^'i- orloD r-b-i Hjq '^1- HrH "^i- H-^ -t?^ 




j; ,-1- t-|ao orw -iN i-^ «^|- cc^4ji r^» Hoo "^1- " • e:Hi 


S 


-1 0*0 "'l- if^D H^ M*o H* -h Hoo ceH< r-jM -♦» e:t* 
— I0c0j>00(»a>05000'-«'!M!«CO 


lO 


-.to ~:|<o .n:© 
c - -•- Hr« H» "*o H* -t?* -I-kHi Hn ^f» 
— iv'J |> 00 00 05 O O r-< ,-. ©q (M CO Tf >o 


o 


(_ oe|« "^.^ ^^ r^ '^•l-"!- wf^ _f^ «J^ ^(_ e-M -tM 0*0 
■- O 00 OS O O — fM 'Tl CO CO 'f o :o r^ 


M^AVOJ 


es.iojj 


1 O C O O O O O O O O 'O 'C' >o o 
f-1 <N CO •^ lO «0 X:- 00 Oi O <M »0 1- O 

1 pH 1— 1 «-H rH <M 



11^ 



126 



Wrought-Irox Shafts, 



I 



I ' 



o 




•sfco 


'-"?= 


-to 


H! 


H=t ^ 


isec 




L-i* 


*fxo 


"51© 


«!f 


»-£ 


:E 




1 


Jr- 


t- 


X 


X 


X 


Oi 


Oi 


Z. 


o 


z. 


^ 


^ 


— 


^ 


O 




Ir-jX 


-*K 


i-,^ 


tfke 


-E 


-P^ 


•-■--c 


-^ 


~i- 


"2^ 








'" 


ir- 


X 


X 


X 


OS 


CS 


C5 


o 


o 


<o 


^ 


^ 


^ 


■M 


o 






«l 




0*0 




r:<c 






OjO 






«)o 


1© 


j,te 














«P0 




-h? Hr' 




-+* 


-4m 




1- 






■"* 


X 


X 


X 


X 


05 


C5 




;z, 


z; 




^ 


^ 


G<I 


5^1 


o 








^-■fs 


^« 


—<: 










tC 


^-:« 


-<= 






o 




ispc 


wee 


-H 






-fK 


-fcl 


t-fcc 


-^ 


1- 






-tN 


Kf^ 




"" 


X 


X 


X 


a 


05 


c 


^ 


X 






^ 


S<J 


3^ 


3<J 








-'fO 








1-"® 


— « 




-E 


~^ 


« 


« 


— ff 


'--(C 


' i^ 




-♦?' 


-1- 




«t«H» 


■i- 


>•>•- 








' 1- 


^1.- 




^' 


_z_ 


X 


X 


C5 




C5 


s 


^ 


^ 




^ 


-M 


5^1 


d 


GO 



<1 
o 



o 
o -^ 

^^ 

P^ 
H 









^ 



" — labc — - -. -. - "'.-, xhr- -fjc -ir^ t-^x -te I— «(>* 
C5C:OiC:C^-^-3<13^'Mr:cCCC 






C "« --i« t_« ^-,« -« ->« t,P= 



o 



t>Pc:f= 



•= Ci O o o f- i-i 3<i 5<i c^: CC CC -r 



^••i-, -;--:-, -+, i^ao «s*o *+D -J- kH -^B -^^ -- 
= O^^CJSOCCCCTJ^rr'* 






',— " — — SM — — — frj Sa— — ft* Wi« wx 

-^1— i^:MC^3CCrtTf'T " 



ir? u-t ?:? o CO 



I- 




^ 


■H 




O 


.= 


i-flo r-H^ tcH* «** ssjx -jM H- esfcc xhji -*c 
— i7^3'MCCCC'^rOiCtC:CX:-X:'Jr-X 




% 


.=' 


3<I3^lCCCC'*«50cC:^J:-J^-XXCi 






H^ c:^ " - oac «»30 «*^ Ofx j>jr *~l- *~:- t-ix -f* 
:ccC'«*'rr^?ctC£-r'XCrsc:riO 


cl 


d 




O 


c 


s4» -t-ix -*,# e>r -JM Kfao le^ -+* i>jr eix -Hr ssfr »-i— 
■^rrOO-^i-XXCiJr-OO— fr-i 


lO 


•H 


kH« -+* ^>^ s» -f* *^ -^ .^^ ^i-. fj- .^ 
i^JtCcCJr-XCiCiO^^C^COCOTf* 


o 


ttie -H» - - Hx -- -^, _^, H^ «to '- *4» r^S» 

X X Ci r; o --' s<j CC TT xo vo O O I:- 

^ ^ — _- -^j 'M 'M -M -N 'M 'M -M 'M <M 


•J^AiOJ 


asjog 


OOiCOOOOOOOOOOO 
(MOi^-OOOi^OOOOOOO 
S0<N(M00C0T:^^u5i0OOIr-i-X 



Teeth of Cast-Iron Wheels. 



127 









TABLE 






Showing the Strength o 


f the Teeth of Cast-iron Wheels at a given 
VelocHy. 








Strength of teeth 


in horse powe 




Pitch of 
teeth in 
inches 


Thick- 
ness of 
teeth in 
inches. 


Breadth 
of teeth 

in 
inches. 


.rat 


3 feet per 
second. 


4 feet per 
second. 


6 feet per 
second. 


8 feet per 
second. 


3*99 


1-9 


7-6 


20-57 


27-43 


41^14 


54-85 


s-'zs 


1-8 


7-2 


17-49 


23^32 


34-98 


46^64 


3-57 


1-7 


6-8 


14-73 


19^65 


29-46 


39^28 


3-36 


1-6 


6-4 


12-28 


16.38 


24-56 


32^74 


3-15 


1-5 


6 


10-12 


13^50 


20-24 


26^98 


2-94 


1-4 


5-6 


8-22 


10^97 


16-44 


21-92 


2-73 


1-3 


5-2 


6-58 


8-78 


13-16 


17-54 


2-52 


1-2 


4-8 


5-18 


6-91 


10-36 


13-81 


2-31 


1-1 


4-4 


3-99 


5^32 


7-98 


10^64 


21 


1-0 


4 


3-00 


4-00 


6-00 


8^00 


1-89 


•9 


3-6 


2-18 


2-91 


4-36 


5^81 


1-68 


•8 


3-2 


1-53 


2^04 


3-06 


3^08 


1-47 


•7 


2-8 


1-027 


r37 


2-04 


2^72 


1-26 


•6 


2-4 


•64 


•86 


1-38 


r84 


1-05 


•5 


2 


•375 


•50 


•75 


1-00 



Furniture Oil. — 1. Linseed oil 1 pint, alkanet ^ oz. Digest in a 
warm place till colored, and strain. 

2. The same, with ^ pint of oil of turpentine. 

3. Linseed oil 1 pint, alkanet root 1 oz., rose pink 1 oz. Let 
them stand in an earthen vessel all night. 

4. A quart of linseed oil, 6 oz. of distilled vinegar, 3 oz. of spirit 
of turpentine, 1 oz. of muriatic acid, and 2 oz. of spirit of wine. i 

5. Linseed oil 8 oz., vinegar 4 oz., oil of turpentine, mucilage, ' 
rectified spirit, each ^ oz. ; butter of antimony J oz. ; muriatic acid 
1 oz. Mix. 

6. Linseed oil 16 oz., black rosin 4 oz., vinegar 4 oz., rectified 
spirit 3 oz., butter of antimony 1 oz,, spirit of salts 2 oz. ; melt the 
rosin, add the oil, take it off the fire, and stir in the vinegar ; let it 
boil for a few minutes, stirring it ; when cool put it into a bottle, 
add the other ingredients, shaking all togetlier. [The last two are 
especially used for reviving French polish.] 

7. Linseed oil 1 pint, oil of turpentine ^ pint, rectified spirit 4 
oz., powdered rosin 1^ oz., rose pink ^ oz. Mix. 

8. Linseed oil 14 oz., vinegar H oz., muriatic acid | oz. Mix. 



128 



Weight of Pipes. 



TABLE 

Showing how to ascertain the weights of Pipes, of various Metals, 
and any diameter required. 



Thick- 
ness in 
parts of 
an inch. 


Wrought 
iron. 


Copper. 




Lead. 


1 

¥ 

To 


•326 
•653 


Hi lbs. plate, 

23i *' 




38 
76 


2 lbs. lead, -483 
4 " " -967 


A 
i 


•976 
1-3 


35 " 

46i " 


1 
1 


14 
52 


8 


" 1-45 
" 1-933 


5 


i • 627 
i'95 


58 '' 
70 " 


1 
2 


9 

28 


11 


" 2-417 
" 2-9 


3^ 


2 277 
2 6 


80i " " 
93 " 


2 

3 


66 
04 


13 
15 


" 3-383 
" 3-867 



Ride. To the interior diameter of the pipe, in inches, add the 
thickness of the metal ; multiply the sum by the decimal numbers 
opposite the required thickness, and under the metals name ; also, 
by the length of the pipe in feet ; and the product is the weight of 
the pipe': ni lbs. 

1. Required the weight of a copper pipe whose interior diameter 
is 7^ inches, its length 6:| feet, and the metal \ of an inch in 
thickness. 

7-5 + -125 r= 7-625 x 1^52 x 6*25 = 72-4 lbs. 

2. What is the weight of a leaden pipe 18^ feet in length, 3 
inches interior diameter, and the metal i of an inch in thickness? 

3 + -25 = 3-25 X 3-867 x 18-5 = 232^5 lbs. 



lb. 



Note. 


—Weight of a cubic inch of 








Lead 


equal 


•4103 




Copper, sheet 




•3225 




Brass, do. 




•3037 




Iron. do. 




•279 




Iron, cast 




•263 




Tin, do. 




•2636 




Znic. do. 




•26 




Water 




•036r 



To SOLDER Tortoise-shell. — Bring the edges of the pieces of shell 
to fit each other, observing to give the same inclination of grain to 
each, then secure them in a piece of paper, and place them between 
hot irons or pincers ; apply pressure, and let them cool. The heat 
must not be so great as to burn the shell, therefore try it first on a 
piece of white paper. 



Weight of Cast-Ikon Balls. 



129 



TABLE 

Of the Weight of Cast-iron Balls, 



Diameter 


Weight 


Diameter 


Weight 
in lbs. 


j Diameter 


Weight 


in inches. 


in l^s. 


in inches. 


! in inches. 


in lbs. 


2 


1-10 


6 


29-72 


10 


137-71 


2i 


1-57 


6i 


83 • 62 


lOi 


148 


28 


2i 


2-15 


H 


37-80 


lOi 


159 


40 


2f 


2-86 


^f 


42-35 


lOf 


171 


05 


3 


8'72 


7 


47-21 


11 


183 


29 


H 


4-71 


n 


62-47 


Hi 


196 


10 


H 


5-80 


H 


58-06 


IH 


209 


43 


Sf 


7-26 


'?f 


64-09 


llf 


223 


40 


4 


8-81 


■ 8 


70-49 


12 


237 


94 


4i 


10-57 


H 


77-32 


12i 


253 


13 


44 


12-55 


8i 


84-56 


12i 


268 


97 


4f 


14-76 


8f 


92-24 


12f 


285 


37 


5 


17-12 


9 


100-39 


13 


302 


41 


5i 


19-93 


9i 


108-98 


13i 


320 


80 


5i 


22-91 


9i 


118-06 


13i 


338 


81 


6f 


26-18 


9f 


127-63 


13f 


357 


93 



1. What will be the weight of a hollow ball or shell of cast-iron, 
the external diameter being 9^, and internal diameter 8f inches if ; 

Opposite ^ are 118-06, and j 

Opposite 8| are 9224, subtract | 



25 82 lbs., weight required. 



2. Requiring to remove a cast-iron ball 37-8 lbs. in weight, and i 
in diameter 6^ inches, and replace it by one of lead of an equal 
weight, what must be the diameter of the leaden ball ? 

Weight of lead to that of cast-iron — 1*56, 

(3.58 

— -— - = L^ 17 6 — 5-6 inches, the diameter. i 

1*06 



Then 



To TRANSFER Engrav[ngs TO Plaster Casts. — Covcr the plate: 
with ink, and polish its surface in the usual way; then put a wall: 
of paper round it, and when completed pour in some finely pow-, 
dered plaster of Paris uiixed in water; jerk the plate repeatedly.' 
to allow the air bubbles to fly upwards, and let it stand one hour;] 
then take the cast off the j)late, and a very perfect impression will' 
be the result ! 



130 






Weight 


OF Flat 


AND Rolled Iron 












1 

o 

ft 
< 

o 

H 

K 

O 
1— ' 

H 
O 


i 

i 

i 

O i 

< ■ 
o 

i 

W 
Q 

b 
< 

ci 

C3 


r^M 


r-< ^ <M 

CN CO Tt< 

o b o 


CO 
CO 

b 










ccH* 


r-( i^- CO 

CO T}H CO 

bob 


-* :0 i- 

Oi Ol O 

b r^ f^ 










^ 


(?^ CO Til 

TT O GO 
h> C. CD 


O GO O 

CM O r-l 

rH r-( CM 


xo OS 
G^ 'M 








rM 


r>^ CO iQ 

lO ir- O 


to r-l CO - 
.— 1 CN (M 


O L- O 
r-l --O S^ 

Ot) CO 4iH 


(M 






1 


O GO r-l 


CO .-1 GO 

j:- CO 00 


CO TtH ^ 

Tt^ o o 


I— 1 xr- 






O O r^ 


1—1 CM CM 


CO rj' tH 


lO O 


^5^ 


CO ^ O 

:0 Cs 3^> 


Cs C<1 uo 

00 o ^ 


00 1-1 rtH 
X- 'cr O 


CO CO 






O' O r-H 


rH CM CO 


CO Tt^ lO 


lO CO 




CO O -t- 
1- ,-H rH 


o ^ r- 
(M c:i CO 


r^ T^ r- 
rr r-l GO 


O lO i:- 
CO CO O 


O 
00 

00 




O^ r^ 


(M 3<l CO 


Tf O lO 


CO J:- 00 


Tj^ O 00 
CJO CM :0 


(M CO O 
lO CO rM 


^ X- (M 
C 00 t- 


CO O rt^ 
O Tti CM 


OO 
O 

b 




O -^ .-H 


(M CO Tj^ 


lO iO CO 


i:- GO Cfs 




Cs rt< CO 


CO GO ^1 
00 i::' £- 


CO 1-1 CO 
CO o o 


O o o; 

lO -^ CO 


Tt< CM CM 

07 M r-t 




O .-< r-^ 


(M CO '^ 


lO CO J:- 


00 Oi CD 


i-H CO lO 


^1 


O O '- 


lO O O 
1-1 ^1 ^1 


gS5 


iO O O 
lO o o 


o o o 

CO ir- c/3 






r-l r-H <M 


CO -* o 


CO i^ 00 


G5 O t-i 


(M ^ CO 


S^ 


lO CO -H 

r-l r- CO 


CO ^1 c- 

^ CO i;- 


ci CO -3H 
05 ::^ CM 


^ss 


CO r- CO 

C» r-i Ti< 


O 

CO 
CM 


,-1 .-! (M 


C^ T^ lO) 


O GO CTi 


O -^ CM 


CO CO 00 


CO 


O Ci 3^ 
(M GO O 


00 ^ O 
J^- O CO 


CO -M 00 

.c OO o 


Tf- O CO 

CO CO CO 


CM ■'^ CO 


o 

(M 
CM 


rH ,^ (M 


CO o o 


I> 00 o 


r-l !M CO 


XO £- O 


CO 


to Ti^ C5 
CO O I-- 


O CO -M 
O' ^ CO 


cr. o 'M 

r^ O C75 


CO O rM 

CM CO O 


00 -^ ^ 

CO r-( CO 


c. CO 

CO £- 


r-( ;M (M 


Th »0 CO 


00 Ci O 


(M CO lO 


O 05 r-l 


I- CM 

-M CO 


1 Hj^ 

1 CO 


£- O ^ 
''t' ^ C5 


r-i 00 lO 

Tti O/ CO 


(M C5 CO 
00 CM j:- 


O O CO 
CM £- r-( 


Tt< OO 'M 
CO lO iO 


O GO 
xt^ (M 


rH (M C<1 


'cH lO -C- 


00 O --1 


CO tH CO 


i:- O CO 
— C^l -M 


Oi vn 

-M CO 


ecH. 
CO 


i— O lO 

lO CO r-l 


(M O 1^ 

J^- CO 00 


vo CM O 
-rf O CO 


O iO 'M 

,-1 L- CO 


O lO o 
C5 O CM 


o o 

iC 00 


^ (M CO 


T^ CO I:- 


05 -H CM 


^ vox:- 


00 -M >0 

r- 'M -M 


CO CO 


^ 


GO ->! «0 
O lO CO 


T^ CM O 

O £- TiH 


00 CO tH 


CM O GO 

r-. GO Tt^ 


00 -rt^ OO 
.— »0 OO 


O CM Ttl 

CO CO O 


.-1 -M CO 


O CO 00 


O rH CO 


VO CO 00 


O CO --o 
(M -M 'M 


CO o t- 
CO Tt^ -^ 


ss 
ssa 


TJd pUB 


j Horr^^H^ 


ecjx H«^ L-:}* 


V^tJfXi 


.-,9:r4* m,o3 


l-t F-t CN 


CM CO CO 











C 


AST 


-Iron Pipes. 








131 




TABLE of the 


Weight of Cast-iron Pipes, in lengths. 






eh 




(J 


^ 
« 


fei 




i \i 


ei i 


g 


JG 


§ 


Weight. 


o 


S 


o 


Weight. 




§ 1 Weight. 


ca 




.J 




M 


H 


h-) 




^ |g 


hJ 1 


Inch. 


In. 


Ft. 


C. qr. lb. 


Inch. 


In 


Ft. 


C. qr. lb. 


Inch. 1 In 


Ft. ; C. qr. lb. 


1 


i 


31 


12 


H 


f 


9 


2 16 


IH 


i 


9 ' 7 




1 


3^ 


21 




i 


9 


2 3 20 




1 


9 ' 6 I 12 


H- 


i 


H 


21 




f 


9 


3 2 21 




1 


9 7 2 8 




f 


H 


1 4 




1 


9 


4 1 21 




1 


9 10 1 2 


2 


i 


6 


1 8 




1 


9 


6 14 


12 


i 


9 5 24 1 




f 


6 


2 


7 


i 


9 


3 7 




1 


9 1 6 2 8 1 


n 


i 


6 


1 16 




i 


9 


3 3 20 




f 


9 17 3 20 




f 


6 


2 10 




f 


9 


4 3 5 




1 


9 ;iO 3 




^ 


6 


3 10 




1 


9 


6 2 4 


]2| 


i 


9 


i 5 1 16 


3 


f 
i 


9 
9 
9 
9 


2 20 
1 6 
1 1 12 
1 3 6 


'71 


i 


9 

9 


3 1 6 

4 22 




% 

i 


9 
9 


i 6 3 9 , 
8 1 






f 
1 


9 
9 


5 10 
7 


13 


1 


9 
9 


11 21 
5 2 20 




f 


9 


2 1 


8 


h 


9 


3 2 4 




i 


9 


7 14 


H 


i 


9 


3 




i 


9 


4 1 25 




f 


9 


8 2 7 




t 


9 


1 21 




i 


9 


5 1 18 




1 


9 


11 2 12 




i 


9 


1 2 14 




1 


9 


7 1 16 


13J 


i 


9 


5 3 7 




-i 


9 


2 8 


H 


i 


9 


3 3 2 




f 


9 


7 1 12 




4 


9 


2 2 




ft 


9 


4 2 26 




f 


9 


8 3 16 


4 


f 

i 


9 
9 


1 1 10 
1 3 12 




1 


9 
9 


5 2 22 
7 3 8 


14 


1 
1 


9 
9 


11 3 24 

6 4 




'B 


9 


2 1 12 


9 


i 


9 


4 




i 


9 


7 2 16 






9 


2 3 21 




5 

8 


9 


5 4 




f 


9 


9 1 


H 


t 


9 


1 2 2 




f 


9 


6 2 




1 


9 


12 1 14 




1 


9 


2 4 




1 


9 


8 26 


14i 


I 


9 


6 24 




f 


9 


2 2 14 


9i 


i 


9 


4 18 




f 


9 


7 3 14 




4 


9 


3 21 




1 


9 


5 1 




f 


9 


9 2 2 


5 




9 

9 


1 2 22 

2 1 10 




4 
1 


9 

9 


6 1 6 

8 2 20 


15 


1 

-1 


9 

9 


12 3 6 
6 1 21 




f 


9 


2 3 n 


10 


i 


9 


4 1 10 




ii 

4 


9 


9 3 7 




f 


9 


3 1 24 




1- 


9 


5 1 26 




1 


9 


13 26 


5^ 


f 


9 


1 3 10 




f 


9 


6 2 14 




li 


9 


16 3 5 




2 


9 


2 2 




1 


9 


9 8 


15i 


1 


9 


6 2 14 




f 


9 


3 18 


10^ 


i 


9 


4 2 14 




f 


9 


10 10 




* 


9 


3 3 7 




f 


9 


5 3 7 




1 


9 


13 2 17 




1 


9 


5 12 




f 


9 


7 




li 


9 


17 1 6 


6 


■1 


9 


2 




1 


9 


9 2 


16 


-J 


9 


7 22 




i 


9 


2 2 21 


11 


i 


9 


4 3 14 




1 


9 


10 1 20 




H 


9 


3 1 17 




f 


9 


6 11 




1 


9 


14 8 




f 


9 


4 16 




f 


9 


7 1 7 




li 


9 


17 3 H 




1 


9 


5 2 20 




1 


9 


9 3 20 




H 


<) 


21 3 4 



132 



Malleable Iron. 



TABLE 

Of the weight of one foot length of Malleable Iron. 



\ SQUARE IRON. 




ROUND IRON. 




1 Scantling 


Weight. 


1 Diameter. 


Weight. 


Ciicumfer. 


Weight. 


\ Inches. 


Pounds. 


Inches. 


Pounds. 


Inches. 


Pounds. 


4- 


0-21 


i 


0-16 


1 


0-26 


f 


0-47 


* 


0-37 


H 


0-41 


1 


0-84 


i 


0-66 


n 


0-59 


1 


1-34 


1 


1-03 


n 


0-82 


i 


1-89 


1 


1-48 


2 


1-05 


i 


2-57 


i 


2-02 


k 


1-34 


1 


8-36 




2-63 


n 


1-65 


: 1* 


4-25 


H 


3-33 


2f 


201 


■ H 


5-25 


H 


4-12 


3 


2-37 


If 


6-35 


H 


4-98 


H 


2-79 


U 


7-56 


H 


5-93 


H 


3-24 


n 


8-87 


H 


6-96 


H 


3-69 


If 


10-29 


If 


8-08 


4 


4-23 


H 


11-81 


H 


9-27 


4i 


5-35 


2 


13-44 


2 


10-55 


5 


6-61 


H 


17-01 


H 


13-35 


5^ 


7-99 


24 


21-00 


n 


16-48 


6 


9-51 


2-1 


25-41 


n 


19-95 


H 


11-18 


o 


30-24 


3 


23-73 


7 


12-96 


u 


41-16 


H 


27-85 


n 


14-78 


4 


53-76 


H 


32-32 


8 


16-92 


U 


68-04 


3f . 


37-09 


H 


19-21 


6 


84-00 


4 


42 21 


9 


21-53 


G 


120-96 


^ 


53-41 


10 


26-48 


7 


164-64 i 

J 


5 


65-93 


12 


31-99 



Fresco Patntsng.— Apply any colors that are not injured by 
lime (according to tq^ste), on a fresh mortared or plastered wall. 

To TAKE Fac-similijs OF SIGNATURES. — Write your name on a piece 
of paper, and while the ink is wet sprinkle over it some finely 
powdered gum arable, then make a rim round it, and pour on it 
some fusible alloy, in a liquid state. Impressions may be taken 
from the plates formed in this way, by means of printing ink and 
; the copperplate pre^. 

WaTCHM.AKEr's On, WHICH NEVER CoRRODES OR TlIICKENS. Take 

olive oil and put h into a bottle, then insert coils of thin sheet 
lead. Expose it to ^he sun for a few weeks, and pour off the clear 
oil. 



Coppers — Cast-Iron Plates. 



133 



TABLE 

Of the Ditne7isions and Weight of Coppers, from 1 to 208 gallons 
The Dimensions taken from lag to brim. 



^ A 


1 




ba . 




c 


1 ts . 




c 


^S 


i ^ 


^ 


.2S 


. 




! -5s 


. 






1 i 


•g.£ 


P 


^ 


iJ 


1^ 


J 


.fa 


o5 




<u 


u o 




►?^ 


u o 




4) 


- 


1 6 


^ 


C*j 


o 


^ 


c — 


o 


> 


n 


1 
1 
1 


U 


24 


15 


22i 


29i 


29 


43i 


m 


2 


3 


24i 
25 


16 
17 


24 

25i 


30 
32 


30 
86 


45 


u 


3 


4i 


54 


15i 


4 


6 


25i 


18 


27 


34 


43 


644- 


16^- 


5 


7i 


26 


19 


28i 


35 


48 


72 


in 


6 


9 


261- 


20 


30 


36 


53 


79i 


184 


1 


lOi 


26f 


21 


31i 


37 


58 


87 


m 


8 


12 


27 


22 


33 


38 


63 


94^ 


20i 


9 


13i 


m 


23 


344- 


39 


67 


lOOi 


21 


10 


15 


21k 


24 


36 


40 


71 


106^ 


2H 


11 


16i 


27f 


25 


374- 


45 


104 


156 


22 


12 


18 i 


28 


26 


39 


50 


146 


219 


22^- 


13 


19^ i 


28^- 


27 


40-i- 


55 


208 


312 


23i 


14 


21 ; 


29 


28 


42 









Weight of Ca^t-Iron Plates, per superficial foot. 
From one-eighth of an inch to one inch thick. 









M inch. 
Ibs. oz. 
19 5f 


% inch. 
Ibs. oz. 
24 2f 


% inch 
Ibs. oz. 
29 


% inch, 
lbs. oz. 
33 13| 




% inch, 
ibs. oz. 
4 13f 


}i inch. 
Ibs. oz. 
9 101 


inch. 
Ibs. oz. 
14 8 


I inch. 
Ibs. oz. 
38 lOf 



The manner of Soldering Ferrules for Tool Handles, <fec. — 
Take your ferrule, lap round the jointing a small piece of brass 
wire, then just wet the ferrule, scatter on the joining ground borax, 
put it on the end of a wire, and hold it in the fire till the brass 
fuses. It will fill up the joining, and form a perfect solder. It 
J may afterwards be turned in the latlie. 

' Cast Engravings. — Take the engraved plate you intend to copy, 
! and arrange a support of suitable materials round it, then pour on 
I it the following allo^- in a state of perfect fusion : tin 1 part ; lead 
I 64 parts; antimony 12 parts. These "cast plates" may be worked 
off on a common printing-press, and offer a ready mode of procuring 
cheap copies of the works of our celebrated artists. 



12 



134 



Cocks — Lead — Lead Pipe — Copper Tubing. 



TABLE 
Of the Bore and Weight of Cocks. 



Content of 
Copper. 



Gallons. 

§0 

50 

80 

120 

150 



Bore of Cock. 



Inches. 

If 

2 

2i 
2i 



Weiaht of 
Cock. 



Pounds. 

7 

8 
12 
19 
26 



Content of 
Copper 



Gallons. 
200 
260 
340 
420 

430 and 
upwards. 



Bore of Cock. 



Inches. 
2f 
3 

3i 
3i 



Weight of 
Cock. 



Pounds. 
30 
34 
44 
56 
70 



Three-fourths of the diameter of the bore, taken at the hinder 
part, will give the diameter of the cock at the mouth. 

TABLE 

Of the Weight of Lead ^ per superficial foot. 

From one -sixteenth of an inch to one inch thick. 



Thickness. 


Weight. 


Thickns. 


Weight. 


Thickns 


Weight. 


Thickness. 


Weight. 


inch. 


lbs. 


inch. 


lbs. 


inch. 


lbs. 


inch. 


lbs. 


l-16th 


3f 


l-8th 


n 


l-4th 


141 


3-4ths. 


44i 


M2th 


5 

■ 


l-6th 


10 


L3rd 


l^i 


1 inch 


59 


1-lOlh 


6 


l-5th 


12 


1-half 


29i 







Weight of Lead Pipe of the usual thicknesses. 
Per foot in length. 



4-inch bore , 


. 1 lb. 1 oz. 






i " 


. 1 lb. 8 oz. 


— 1 lb. 12 oz. — 2 lbs. 




1 


. 2 lbs. 


— 2 lbs. 11 oz. — 2 lbs. 


14 oz. 


li " 


. 3 lbs. 


-— 3 lbs. 11 oz. — 4 lbs. 


7 oz. 


n " 


. 4 lbs. 


— 4 lbs. 11 oz. — 5 lbs. 


9 oz. 


2 


. 5 lbs. 9 oz. 


_ 7 lbs. ~ 8 lbs. 


5 oz. 


2i " 


. 7 lbs. 


— 8 lbs. 9 oz. —10 lbs. 





Weight of Copper Tithing. 
Of the usual thickness. 
When the inside diameter is J of an inch, 3 ounces ; f of an 
inch, 5 ounces ; ^ of an inch, 6 ounces ; f of an inch, 8 ounces ; 
and f of an inch, 10 ounces per foot. 



Strength of Materials. 



135 



STRENGTH OF MATERIALS. 

Materia. Is of constriiction are liable to four different kinds of 
strain, viz., stretching, crushing, transverse action, and torsion or 
twisting: the first of which depends upon the body's tenacity 
alone ; the second, on its resistance to compression ; the third on 
its tenacity and compression combined; and the fourth, on that 
property by which it opposcc- any acting force tending to change 
from a straight line, to that of a spiral direction, the fibres of which 
the body is composed. 

In bodies, the power of tenacity and resistance to compression, in 
the direction of their length is as the cross-section of their area 
multiplied by the results of experiments on similar bodies, as exhi- 
bited in the following table : 

TABLE 

Showing the Tenacitie.% Resistances to Coinpression, and other Pro- 
perties of the comino7i Materials of Construction. 



Names of Bodies. 



Asli 

Beech 

Brass 

Brick 

Cast iron 

Copper (wrought). . . 

Elm 

Fir, or Pine, white . . 

red 

" '* yellow. 

Granite 

Gun-metal (copper ) 
8, and tin 1 . . . . \ 

Malleable iron 

Larch 

Lead 

Mahogany,Honduras 

Marble 

Oak 

Rope (1 in. in eircum.) 

Steel 

Tin (cast) 

Zinc (sheet) 



Absolute. 



Tenacity 
in lbs. 
per sq. 
inch. 



14130 
12225 
17968 
275 
18434 
33000 
9720 
12346 
11800 
11835 

35838 

56000 

12240 

1824 

11475 

551 

11880 

200 

128000 

4786 

9120 



Resistance 
loconnpres- 
siun '\v\ lbs 
per sq. in. 



8548 

10304 

562 

86397 

1033 
2028 
5375 
5445 
10910 



5568 

8000 
6060 
9504 



Compared with Cast Iron. 



Its 

strength 



23 
0-15 
0-435 

1-000 

0-21 
0-23 
0-3 
0-25 

0-65 

1.12 
0136 
0-096 
0-24 

0-25 



0-182 
0-365 



Its 
extensi- 
bility 



2-6 
2-1 
0-9 

10 

2-9 
2-4 
2-4 
2-9 

1-25 

0-86 

2-3 

2-5 

2-9 

2-8 



0-75 
05 



Its 
stiffness 



0.089 
0-073 
0-49 

1-000 

0-073 

0-1 

0-1 

0-087 

0-535 

1-3 

0-058 

0-0385 

0-487 

0-093 



0-25 
0-76 



136 



Ropes and Chains — Timber. 



TABLE 

Of the Comparative Strength and Weight of Ropes and Chains. 



<u 








c 


a 








c 


.*" ^ 


«-^ 




-2 


£5 


2S 


Sj-o 


^1 


!ni 




o-^ 


i = 


«.- 


!^ 


£ = 


c^ 




S3.S 


i= 


£-H 




■5s 






if 


r 

5 


■Sh 




^^1 


Is 


3i 


2i 


r% 


H 


1 oi 


10 


23 


1 


43 


10 


4i 


4i 


f 


8 


1 16f 


lOf 


28 


1-1 


49 


11 11 





of 


7 


101 


2 10 


Hi 


304 


I in. 


56 


13 8 


54 


7 


i 


U 


3 0^ 


12i 


36 


IfV 


63 


14 18 


6i 


9f 


9 


18 


4 3i 


13 


39 


li 


Tl 


16 14 


7 


ni- 


f 


22 


5 2 


13f 


45 


3 
16^ 


79 


18 11 


8 


ls 


Ta 


2*7 


6 4^ 


14i 


48i 


n 


87 


20 8 


8f 


19 


f 


82 


V 7 


15i 


56 


ii% 


96 


22 13 


n 


21 


13 
IB" 


37 


8 13^ 


16 


60 


If 


106 


24 18 



Note.—\l must be understood, and also borne in mind, that in estimating the amount 
of tensile stiain to which a body is subjected, the weight of the body itself must also be 
taken into account ; for according to Us position so may it approximate to its whole 
weight, in tending to produce extension within itself; as in the almost constant applica- 
tion of ropes and chains to great depths, considerable heights, &c. 

Resistance to L<ateral Pressure, or Transverse 
Action* 

TABLE 

Of Data, containing the Results of Experiments on the Elasticity 
and Strength of various Species of Timber. 



Species of 


Value 


Value 


Species of 


Value 


Value 


Timber. 


ofE. 


of S. j 


Timber. 


olE. 


ofS. 


Teak, . . . 


174-7 


2462 


Ehn, . . . 


50-64 


1013 


Poona, . . . 


122-26 


2221 


Pitch pine, . 


88-68 


1632 


English Oak, . 


105 


1672 


Red pine, . . 


133 


1341 


Canadian do., . 


155-5 


1766 


New Eng. fir, 


158-5 


1102 


Dantzic do., . 


86-2 


1457 


Riga do., . . 


90 


1100 


Adriatic do., . 


70-5 


1383 


Mar Forest do. 


63 


1200 


Ash, .... 


119 


2026 


Larch, , . . 


76 


900 


Beech, . . . 


98 


1556 


Norwayspruce, 


105-47 


1474 



Strength of Matp:rials. 1B^ 



The strength of a square or rectar.gular beam to resist lateral 
pressure, acting in a perpendicular direction to its length, is as the 
breadth and square of the depth, and inversely as the length. Thus, 
a beam twice the breadth of another, all other circumstances 
being alike, equals twice the strength of the other ; or twice the 
depth, equal four times the strength, and twice the length, equal 
only half the strength, <fec., according to the rule. 

To find the dimerkHion^ of a beam capable of maintaining a given 
zveig/Uj with a given degree of deflection^ when supported at both ends. 

Rule. Multiply the weight to be supported in lbs. by the cube of 
the length in feet ; divide the product by 32 times the tabular 
value of E, multiplied into the given deflection in inches; and the 
quotient is the breadth multiplied by the cube of the depth in 
inches. 

Note 1.— When the beam is intended to be square, then the fourth root of the quotient 
is the breadth and depth required. 

Note 2.— If the beam i.s to be cylindrical, multiply the quotient by 17, and the fourth 
root of the product is the diameter. 

Example. The distance between the supports of a beam of Riga 
fir is 16 feet, and the v^^eight it must be capable of sustaining in the 
middle of its length is 8000 lbs., with a deflection of not more than 
f of an inch ; what must be the depth of the beam, supposing the 
breadtli 8 inches? 



16 X 8000 
90 X 32 X -75 



= 15175 -T- 8 = Visy? = 12-35 in., the depth. 



To determine the absolute strength of a lectangidar beam of tiinber, 
when supported at both ends, and loaded in the middle of its length, as 
beams in general ought to be calculated to, so that tJteyinay be rotdered' 
capable of %vithstanding all accidental casea of emergency. j 

Rule. Multiply the tubular value of S by four tirnes the depth of 
the beam in inches, and by the area of the cross section in inches;- 
divide the product by the distance between the supports in inches, j 
and the quotient will be the absolute strength of the beam in lbs. 

No^^ 1.— If the beam be not laid horizontally, the di.stance between the .supports, forj 
calculation, must be the horizontal distance. 

Note 2.— One fourth of J he weight obtained by the rule is the {greatest weight thai | 
ought to be applied in practice as permanent load. i 

Note. 3. — If the load is to be ap[)lied at any other point than the middle, then thai 
st.enpth will be a.s the r>roduct of the two distances is to the square of half the leii^rth of 
the beam between the supports ; or. twice the distance from one ii\n\~ multiplied by twice | 
from the other, and divided by the whole length, ecpial the etVective length ot the beam. | 

Example. In a building 18 feet in width, an engine boiler of 5| 
tons is to be fixed, the centre of which is to be 7 feet from the wall; 
and having two pieces of red pine, 10 inches by 6, which I can lay 
across the two walls for the purpose of slinging it at each end, may 
I with sufficient confidence apply them, so as to etfect this object? 

i'2* ^ 



138 Strength of Materials. 



2240 X 5-5 ^,,^ ,v . ,1 

= 6160 lbs. to carry at each end. 



4 

And 18 feet — 7 = 11, double each, or 14 and 22, then 
14 X 22 



= 17 feet, or 204 inches, effective length of beam. 



13 

rr. . . , . , . 1341 X 4 X 10 X 60 ^^ „ 

Tabular value of S, red pine, =: ,y . = 15776 lbs. 

the absolute strength of each piece of timber at that point. 

To determine the dimensions of a rectang^dar beam capable of sup- 
porting a required weight, with a given degree of dejlection, when fixed 
at one end. 

Rule. Divide the weight to be supported, in lbs., by the tabular 
value of E, multiplied by the breadth aad deflection, both in 
inches ; and the cube root of the quotient, multiplied by the 
length in feet, equal the depth required in inches. 

Example. A beam of ash is intended to bear a load of 700 lbs. at 
its extremity, its length being 5 feet, its breadth 4 inches, and the 
deflection not to exceed ^ of an inch. 

Tabular value of E = 119 x 4 x -5 = 238 the divisor ; 

then 700 -f- 238 = v 294 x 5 = 725 inches, depth of the beam. 

To find the absolute strength of a rectangidar beam, lohen fixed at 
one C7id and loaded at the other. 

Rule. Multiply the value of S by the depth of the beam, and by 
the area of its section, both in inches : divide the product by the 
leverage in inches, and the quotient equal the absolute strength of 
the beam in lbs. 

Example. A beam cf Riga >ir, 12 inches by 4^, and projecting 
6^ feet from the wall ; what is the greatest weight it will support 
at the extremity of its length ? 

Tabular value of S = 1100 
12 x 4*5 = 54 sectional area, 

^ 1100 X 12 X 54 

Then, -—- = 0138-4 lbs. 

78 

AV hen fracture of a beam is produced by vertical pressure, the 
fibres of the lower section of fracture ars separated by extension, 
whilst at the same time those of the upper portion are destroyed by 
compression ; hence exists a point in section where neither the one 
nor the other takes place, and which is distinguished as the point 



Strength of Materials. 139 



of neutral axis. Therefore, by the law of fracture thus established, 
and proper data of tenacity and compression given, as in the table 
(p. 135), we are enabled to form metal beams of strongest section 
with the least possible material. Thus, in cast iron, the resistance 
to compression is nearly as 6^ to 1 of tenacity ; consequently 
a beam of cast iron, to be of strongest section, must be of the follow- 
ing form, and a parabola in the direction of its length, 
the quantity of material in the bottom flange being 
about 6^ times that of the upper. But such is not the 
case with beams of timber ; for although the tenacit}' 
of timber be on an average twice that of its resistance 
to compression, its flexibility is so great that any considerable 
length of beam, where columns cannot be situated to its support, 
requires to be strengthened or trussed by iron rods, as in the follow- 
ing manner : 





and these applications of principle not only tend to diminish deflec- 
tion, but the required purpose is also more effectively attained, and 
that by lighter pieces of timber. 

To ascertain the absolute strength of a cast-iron beam of the pre- 
ceding forrn, or that of strongest section. 

Rule. Multiply the sectional area of the bottom flange in inches 
by the depth of the beam in inches, and divide the product by the 
distance between the supports, also in inches ; and 514 times the 
quotient equal the absolute strength of the beam in cwts. 

The strongest form in which any given quantity of matter can be 
disposed is that of a hollow cylinder ; and it has been demonstrated 
that the maximum of strengtli is obtained in cast iron when the 
thickness of the annulus or ring amounts to ^th of the cylinder's 
external diameter ; the relative strength of a solid to that of a hol- 
low cylinder being as the diameters of their sections. 

ToRToisE-siiELL c^ouxD FOR Metal. — Covcr thc platcs intended 
to represent the transparent parts of the tortoise-shell with a thin 
coat of vermilion in seed-lac varnish. Then brush over the whole 
with a varnish composed of linseed oil boiled with umber until it is 
almost black. The varnish may be thinned with oil of turpentine 
before it is used. When the work is done it may be set m an 
oven, with the same precautions as the black varnish 

FoRdE IN Pile-Driving. — In a sandy soil the greatest force of a 
pile-driver will not drive a pile over hfteen feet. 



140 



Strength oi< ^i.»-.=r.». 



TABLE 

Showing the Weight or Pressure a Beam of Cast Iron, 1 inch in 
breadth, will sustain, without destroying its elastic force, when it is 
supported at each end, and loaded t?i tJie middle of its length, and 

'^^ho the deflect' on in the niiddle which that weight icill produce. 



1 

Length. 

'il 


6 feet. 


7 leet. 


8 feet. 


9 feet. 


10 feet. 


Si 


Mi 
II 


t£J3 

r 


a "5 


ii 
1 


S en 

II 
<S-5 


ii 


11 


.s 1 §«• 

$ Si 


3 


1278 


•24 


1089 


•33 


954 


•426 


855 


•54 


765 


•66 


H 


1789 


•205 


1482 


•28 


1298 


•365 


1164 


•46 


1041 


•57 


4 


2272 


•18 


1936 


•245 


1700 


•82 


1520 


•405 


1360 


•5 


^ 


2875 


•16 


2450 


•217 


2146 


•284 


1924 


•36 


1721 


•443 


5 


3560 


•144 


3050 


•196 


2650 


•256 


2375 


•32 


2125 


•4 


6 


5112 


•12 


4356 


•163 


3816 


•213 


3420 


•27 


3060 


•33 


1 


6958 


•103 


5929 


•14 


5194 


•183 


4655 


•23 


4165 


•29 


8 


9088 


•09 


7744 


•123 


6784 


•16 


6080 


•203 


5440i ^25 


9 






9801 


•109 


8586 


•142 


7695 


•18 


6885 -22 


10 






12100 


•098 


10600 


•128 


9500 


•162 


8500j '2 


1 ^^ 










12826 


•117 


11495 


•15 


10285' -182 


12 










15264 


•107 


13680 


•135 


12240 -17 


13 














16100 


•125 


14400 •I 54 


14 
6 














18600 


•115 


16700 -143 


12 feet. 


14 feet. 


16 feet. 


18 fe 


et. 


20 feet. 


2548 


•48 


2184 


•65 


1912 


•85 


1699 


1-08 


1530 


1-34 


1 


347] 


•41 


2975 


•58 


2603 


•73 


2314 


•93 


2082 


114 


8 


4532 


•36 


3884 


•49 


3396 


•64 


3020 


•81 


2720 


100 


9 


573c 


•32 


4914 


•44 


4302 


•57 


3825 


•72 


34S8 


•89 


10 


7088 


•28 


6071 


•39 


5312 


•51 


4722 


•64 


4250 


•8 


11 


857C 


•26 


7346 


•36 


6428 


•47 


5714 


•59 


5142 


•73 


12 


10192 


•24 


8736 


•33 


7648 


•43 


6796 


•54 


6120 


•67 


13 


11971 


•22 


10260 


•31 


8978 


.•39 


7980 


•49 


7182 


•61 


14 


13883 


•21 


11900 


•28 


10412 


•36 


9255 


•46 


8330 


•57 


15 


15937 


•19 


13660 


•26 


11952 


•34 


10624 


'•43 


9562 


•53 


16 


18128 


•18 


15536 


•24 


13584 


,•32 


1 12080 


•40 


10880 


•5 


17 


'2050(J 


•17 


17500 


•23 


15358 


'o 


13647 


•38 


12282 


•47 


18 


i2293i: 

i 


•16 


19656 


•21 


17208 


|-2S 


; 157 0.0 


•36 


13752 


•44 



No/e.— This table shows the greatest weight that ever uugiit to be laid upon a beam 
for permanent load : and, if there be any liability to jerks. &c., ami>ie allowance must be 
made ; also, the weight of the beam itself must be included. 



Strength of Materials. 141 



To find the weight of a cast-iron beam of given dimensions. 

Ride. Multiply the sectional area in inches by the length in 
feet, and by 3*2, the product equal the weight in lbs. 
. Ex. Required the weight of a uniform rectangular beam of cast 
iron, 16 feet in length, 11 inches in breadth, and 1^- inch in thick- 
ness. 

11 X 1-5 X 16 X 3-2 = 844-8 lbs. 

Resistance of Bodies to Flexure by Vertical Pressure. 

When a piece of timber is employed as a column or support, its 
tendency to yielding by compression is different according to the 
proportion between its length and area of its cross section ; and 
supposing the form that of a cylinder whose length is less than 
seven or eight times its diameter, it is impossible to bend it by any 
force applied longitudinally, as it will be destroyed b}^ splitting 
before that bending can take place ; but when the length exceeds 
this, the column will bend under a certain load, and be ultimately 
destroyed by a similar kind of action to that which has place in 
the transverse strain. 

Columns of cast iron, and of other bodies, are also similarly cir- 
cumstanced, this law having recently been fully developed by the 
experiments of Mr. Hodgkinson on columns of different diameters, 
and of different lengths. 

When the length of a cast-iron column with flat ends equals 
about thirty times its diameter, fracture will be produced wholly 
by bending of the material. When of less length, fracture takes 
place partly by crushing and partly by bending. But, when the 
column is enlarged in the middle of its length from one and a half 
to twice its diameter at the ends, by being cast hollow, the strength 
is greater by \th. than in a solid column containing the same 
quantity of material. 

To determine the dimeiisions of a .support or column to bear^ 
ivithotU sensible curvature, a given pressure in the direction of its 
axis. 

Rule. — Multiply the pressure to be supported in lbs. by the 
square of the column's length in feet, and divide the product by 
twenty times the tabular value of E ; and the quotient will be 
equal to the breadth multiplied by the cube of the least thickness, 
both being expressed in inches. 

yote I.— When the pillar or support is a square, its side will be the fourth root of the 
quotient. 

'2 If the pillar or column be a cylinder, multiply the tabular value of E by 12, and the 
K.urth root of the quotient equal the diameter. 

JiJx. I. What should be the least dimensions of an oak support, 
to bear a weight of 2240 lbs , without sensible flexure, its breadth 
being 3 inches, and its length 5 feet ? 



142 



Strength of Materials. 



Tabular value of E = 105, 

, 2240 X 5' 
and 



20 X 105 X 8 



= ^8-888 = 2-05 inches. 



jEx. 2. Required the side of a square piece of Riga fir, 9 feet in 
length, to bear a permanent weight of 6000 lbs. 
Tabular value of E = 96, 

, 6000 X 9- , -- ^ . , , 

and • X </lo6 ■=. 4 inches nearly. 

20 X 96 ^ ^ 



TABLE 

Of the Dhnensio7is of Cyli7idrlcal Columns of Cast Iron to siistain a 
given load or pressure ivith safety. 



si 


LEXGTn OR Height in Feet. 




4 


6 


8 10 12 


14 


16 


18 


20 


22 


24 




Weight or Load in Cwts. 



2 
2i 



4 

5 
' 6 

n 

8 

9 

10 

11 

12 



72 

119 

178 

247 

326 

418 

522 

607 

1032 

1333 

1716 

2119 

2570 

3050 



60 


49 


40 


32 


26 


22 


18 


15 


13 


105 


91 


77 


65 


55 


47 


40 


34 


29 


163 


145 


128 


111 


97 


84 


73 


64 


66 


232 


214 


191 


172 


156 


135 


119 


106 


94 


310 


288 


266 


242 


220 


198 


178 


160 


144 


400 


379 


351 


327 


301 


275 


251 


22f^ 


208 


501 


479 


452 


427 


394 


365 


337 


31(^ 


285 


592 


573 


550 


525 


497 


469 


440 


413 


386 


1013 


989 


959 


924 


887 


848 


808 


765 


725 


1315 


1289 


L259 


1224 


1185 


1142 


1097 


105'i 


U)05 


1697 


1672 


1610 


1603 


1561 


1515 


1467 


1416 


1364 


2100 


2077 2045 


2007 


1964 


1916 


1865 


1811 


1755 


2550 


2520 


2490 


2450 


2410 


2358 


2305 


2248 


2189 


3040 


3020 


2970 


2930 


2900 


2830 


•2780 


2730 


2670 



11 

25 

49 

83 

130 

189 

262 

360 

686 

959 

1311 

1697 

2127 

2600 



Practiced Utility of the preceding Table. 

Ex. Wanting to support the front of a building with cast-iron 
columns 18 feet in length, 8 inches in diameter, and the metal 1 
inch in thickness; what weight may I confidently expeet each 
column capable of supporting without tendency to deflection ? 



Strength of Materials. 143 



Opposite 8 inches diameter and under 18 feet = 1097 
Also opposite 6 in. diameter and under 18 feet = 440 



Note,— The strength of cast iron as a column being 10000 

steel *' =2-518 

" wrought iron " = 1745 

" (oak) Dantzic " rr -1088 

" red deal " = "0785 

Elasticity of Torsion^ or Resistance of Bodies to Twisting, 

The angle of flexure by torsion is as the length and extensibility 
of the body directly, and inversely as the diameter; hence the 
length of a bar or shaft being given, the power, and the leverage 
the pov/er acts with, being known, and also the number of degrees 
of torsion that will not affect the action of the machine, to deter- 
mine the diameter in cast iron, with a given angle of flexure. 

Rale. Multiply the power in lbs. by the length of the shaft in 
feet, and by the leverage in feet; divide the product by fifty-five 
times the number of degrees in the angle of torsion ; and the fourth 
root of the quotient equal the shaft's diameter in inches. 

Ex. Required the diameters for a series of shafts 35 feet in 
length, and to transmit a power equal to 1245 lbs., acting at the 
circumference of a wheel 2^ feet radius, so that the twist of the 
shafts on the application of the power may not exceed one degree. 

1245 X .35 X 2-5 , . . ,. 

■ — ; ;; • = -v/l'^Sl = G'S'/ inches in diameter. 

6o X 1 ^ 

Relative Strength of Metals to resist Torsion. 
Cast iron . . . =1* I Swedish bar iron . =1*05 
Copper .... — -48 I English do. . . =1-12 
Yellow Brass . . — -511 I Shear steel . . . =1-96 
Gun-metal ... = -55 Cast do =2*1 



Map Colors. 

YELLOW. 

1. Dissolve gamboge in water. 

2. Make a decoction of French berries, strain, and add a little 
gum arabic. 

RED. 

1. Make a decoction of Brazil dust in vinegar, and add a little 
gum and alum. 

2. Make an infusion of cochineal, and add a little gum. 

BLUE. 

A weak mixture of sulphate of indigo and water, to which add a 
little gum. 

GREEN. 

1. Dissolve crystals of verdigris in water, and add a little gum. 

2. Dissolve sap green in water, and add gum. 



144 



Weight of Iron, Copper, and Bkass. 



TABLE 

Of the Weight of a Superficial Foot of Plate or Sheet Iran, Copper, 
and BrasSf in pounds. 



cs 

a* 

C 











ron. 




1 

32 


i 1-25 


■ 


1 


2-5 




i 


o 


'i. 


3 


'7 • '^ 




16 


< 




i 


10 


1 ? 






^ 


12*5 


■l 


1 


15 


: 2 


^ 


17-5 


i^ 


i 


20 


s 


tHt 


22-5 


44 


1 


25 


in 


u 


27*5 




i 


30 




i 


35 


■^vn 


1 


40 





Iron. 



1 I 12-5 

2 I 12 
11 
10 



4 ; 



Copper. 


Brass. 


14-5 


13-75^ 


13-9 


13-2 


12-5 


12-1 j 


11'6 


11 i 



5 i 

6 

7 

8 

9 
10 
11 
12 
13 



8-74i 10-1 



8-12 
7-5 
6-86 
6-24 
5*62 
5 

4-38 

3-75 

iM i i3^'12 



15 



2-82 



9-4 
8-7 
7-9 
7-2 

6'5 

5-8 

5-08 

4-34 

3-6 

3-27 



fee 

;! fee 

61 i| © 
■93;!*? 

•25;|| 

86:! i 

18|| 

5 • 

81 

12 

43 

1 I 



No 


Iron. 


Copr. 


Brass. 


16 


2*6 


2-9 


2*76 


17 


2-18 


2-52 


2-4 


18 


1-86 


2-15 


2-04 


19 


I'l 


1-97 


1-87 


20 


1-54 


1-78 


1-69 


21 


1-4 


1-62 


1-54 


22 


1«25 


1-46 


137 


28 


1-12 


1-3 


1-2S 


24 


1 


1-16 


ri 


25 


•9 


1-04 


•99 


26 


•8 


•92 


•88 


27 


'1% 


•83 


•79 


28 


•64 


•74 


•7 


29 


'56 


•64 


•61 


30 


•5 


•58 


•55 



2\,^oie. — Xo. 1 wire gauge equal ^ths of an inch. 



'^ 11 
" 16 
« 00 



A 



1 

32 



Tke great Taxiety of thicknesses into which copper is manufac- , 
tured, cause in trade the weight to be named whereby to deter- j 
mine the thickness required, "^the unit being that of a common j 
sheet, so designated, viz. 4 feet by 2 feet, in lbs., thus : 

A 70 lb. plate is ^ths "of an inch in thickness 

.. 46JL " ^ 



*^ lU 

'' 6 



3,3 

9r 



Wkight of Materials. 



14.5 



The thickness of lead is also in common determined or under- 
stood by the weight, the unit being that of a square or superficial 
foot; thus: 

ckness. 
6 '^ A " 



n 
11 

15 



Coynparative Weights of Different Bodies. 

Bar iron being J, 
Cast iron = '95 
Steel = 1-02 

Copper = 1'16 
Brass — 1*09 

Lead = 1*48 



Cast iron being 1, 
Bar iron = 1*0 
Steel = 1-08 

Brass z=z 1-16 
Copper = 1-21 
Lead := 1'56 



1. Suppose I have an article of plate iron, the weight of which 
is 728 lbs., but want the same of copper, and of similar dimensions, 
what will be its weight ? 

728 X 1-16 = 844-48 lbs. 

2. A model of di'J pine, weighing 32^ lbs,, and in w^hich the iron 
for its construction forms no material portion of the Aveight, what 
may I anticipate its weight to be in cast iron ? 

82-5 X 16 = 520 lbs. 

Note — It frequently occurs, in the formation or construction of models, that neither the 
quality nor condition of the timber can be properly estimated ; and, in such cases, it 
may be a near enough approximation to reckon 15 lbs. of cast iron to each lb. of model. 



Silvering Powder, <fec., for silvering copper, covering the worn 
parts of plated goods, &c. — 1. Nitrate of silver 30 gr., common salt 
30 gr., cream of tartar 3^ dr. Mix. Moistened with water, and 
rubbed on dial plates or other copper articles, it coats them with 
silver. 

2. Silver precipitated from its nitric solution by copper 20 gr., 
alum 30 gr., cream of tartar 2 dr., salt 2 dr. 

3. Precipitated silver I- oz , common salt 2 oz., muriate of am- 
monia 2 oz., corrosive sublimate 1 dr. Make it into a paste with 
water. Copper utensils are previously boiled with tartar and 
alum, and rubbed with this paste, then made red hot, and after- 
wards polished. 

4. Dissolve muriate of silver in a solution of hyposulphite of 
soda, and mix this with prepared hartshorn, or other suitable 
powder. 

Platixa for Springs. — Platinum 1 part; gold 12 parts. Add 
the platinum to the gold in a state of fusion. 



18 



146 



Mensuration of Timber, 



Table§ by which to facilitate the I?Iensnra- 
tion of Timber. 

1. Flat or Board Measure. 



Breadth 


Area of a 


Breadth 


Area of a 


! Breadth 


Area of a 


ia inches. 


.'meal toot, i 


in inches. 


lineal foot. 


in itlcheg. 


lineal foot. 


i 


•0208 1 


4 


•3334 


i 

8 


•6667 


i 


•0417 


4i 


•3542 


' 8i 


•6875 


f 


•0625 


44 


•375 i 


\ H 


•7084 


1 


•0834 : 


4i 


•3958 


i 8f 


•7292 


li 


• 1042 


5 


•4167 


i 9 


•75 


H 


•125 


5i 


•4375 ' 


i n 


•7708 


If 


•1459 


5i 


•4583 


\ ^* 


••7917 


2 


•1667 


of 


•4792 


1 9f 


•8125 


H 


•1875 : 


6 


•5 


1 10 


•8334 


2i 


•2084 ' 


6i 


•5208 


i m 


•8542 


2* 


•2292 


6i 


•5416 


i 104 


•875 


3 


•25 


6f 


•5625 


1 lOf 


•8959 


H 


•2708 \ 


7 


•5833 


! 11 


•9167 


H 


•2916 


Vi 


•6042 


1 Hi 


•9375 


3t 


•3125 


H 


•625 


\ IH 


•9583 






7i 


• 6458 


i iif 

i 


•9792 



J^^' Application and Use of the Table. 

1. Required the number of square feet in a board or plank 16-^ 
feet in length, and 9f inches in breadth. 

Opposite 9f is -8125 x 16^5 = 13^4 square feet 

2. A board 1 foot 2f inches in breadth, and 21 feet in length ; 
T^hat is its superficial content in square feet ? 

Opposite 2f is "2292, to which add the 1 foot. 
Then 1-2292 x 21 == 25 '8 square feet 

3. In a board 15| inches at one end, 9 inches at the other, and 
14-|- feet in length, how many squai\e £eet,^ 

15^5 + 9 

= 12J, or 1-0208; and 1'0208 x 14^5 = 14*8 square feet 



To GIVE Iron a temper to <:ut Porphyry. ^=r-Make your iron red 
hot, and plunge it into distilled water from nettles, acanthus, and. 
pilosella, or in the very juice pounded out from these plants. 

Paste for cleaning Metals. — ^Take oxalic acid 1 part; rotten- 
stone 6 parts. Mix with equal parts of .train .oil and spirits of 

turpentine to a paste. 



Mensjjration of Timbkr. 



141 



2. Cubic or Solid Measure. 


Mean 'A Cu 


bic feet 


Mean U 


Cubic feet 


Mean H 


Cubic feet 


girth in ir 


each 


pinYt id 


in each 


girth in 


in each 


inches. line 


al foot. 


inches. 


lineal foot. 


inches. 


lineal foot. 


6 


25 


14 


1-361 


22 


3-362 


6i 


272 


14i 


1-41 


221 


3-438 


H 


294 


144- 


1-46 


22i 


3-516 


6f 


317 


14f 


1-511 


22| 


3 • 598 


7 


340 


15 


1-562 


23 


3-673 


U 


364 


15i 


1-615 


. 2H 


3-754 


n 


39 


1-H 


1 668 


23i 


3-835 


H 


417 


15f 


1-722 


234- 


3-017 


8 


444 


]6 


1-777 


24 


4- 


8i 


472 


16i 


1-8:^3 


241 


4-084 ; 


81 


501 


16i 


1-89 


, 244- 


4-368 


8f 


531 


16f 


1 • 948 


, 241 


4-254 


9 


562 


17 


2-006 


25 


4-34 


9i 


594 


m 


2-066 


i 251 


4-428 


91 


626 


17i 


2-126 


25i 


4-516 


91 


659 


17f 


2^187 


, 25f 


4-605 


10 


694 


18 


2-25 


i 26 


4-694 


lOi 


73 


m 


2-313 


■ 261 


4-7S5 


10-1 


766 


m 


2-376 


'' 264 


4 S76 


lOf 


803 


18f 


2-442 


2()f 


4 •96:) 


11 


•84 


19 


2-506 


i ^^ 


5 062 


m 


878 


19i 


2-574 


271 


5-158 


ni 


•918 


19i 


2-64 


■ 'All 


5-252 


iif 


•959 


]9f 


2-709 


271 


5-348 


12 1 




20 


2-777 


t 28 


5-444 


12i 1 


•042 


201 


2 -898 


281 


5-542 


12i 1 


•085 


20i 


2-917 


28^ 


5-64 


13f 1 


•129 


20| 


2-99 


' 28f 


5-74 


13 1 


•174 


21 


3 062 


29 


5-84 


13i 1 


•219 


21i 


8-136 


29i 


5-941 


13i 1 


265 


2H 


3-209 


29i 


6-044 


13f 1 


•313 


21f 


3-285 


29f 


6-146 



In the cubic estimation of timber, custom has established the rule 
of i the mean girt being the side of the square considered as the 
cross sectional dimensions; hence, multiply the number of cubic 
feet per lineal foot, as in the Table of Cubic Measure, opposite the 
^ girth, and the product is the solidity of the given dimensions in 
cubic feet. 

Suppose the mean J girth of a tree 21 J inches, and its length 16 
feet, what are its contents in cubic feet? 

3-136 X 16 = 50-176 cubic feet. 



1-48 



C Y LiNDERs — Pipes. 



CAST METAL CYLINDERS. 
The Cylinders are solid, each 1 foot in length. 



Diameter. 


Iron. 


Copper. 


Brass. 


Lead. 


inches. 


lbs. 


lbs. 


Jbs. 


lbs. 


1 


2-5 


3 





2 


•9 


3-9 


2 


9-8 


12 





11 


-4 


15-5 


3 


22-1 


27 





25 


•8 


34-8 


4 


39-3 


47 


9 


45 


8 


61-9 


5 


61-4 


74 


9 


71 


6 


96-7 


6 


88-4 


107 


8 


103 





139-3 


7 


120-3 


146 


8 


140 


2 


189-6 


8 


157*1 


191 


7 


183 


2 


247-7 


9 


198-8 


242 


7 


231 


8 


313-4 


10 


245-4 


299-5 


286-2 


387-0 



CAST-IRON" PIPES. 

Table showing the Weight of Pipes 1 foot long, of bores from 1 inch 
to 12 inches in diameter, advancing by ^ of an inch ; and of thick- 
nesses from i of an inch to 1^ inches, advancing by ^ of an inch. 



bore. 


i 


t 


■ 


1- 


f 


i 


1 


H 


li 


in. 


lbs. 


lbs. 


lbs. 


Ibs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


1 


3-1 


5-1 


7-4 


10-0 


12-9 


16-1 


19-6 


23-5 


27-6 


li 


3-7 


6-0 


8-6 


11-5 


14-7 


18-3 


22-1 


26-2 


30-7 


H 


4-3 


6-9 


9-8 


13-0 


16-6 


20-4 


24-5 


29-0 


33 7 


If 


4-9 


7-8 


11-1 


14-6 


18-4 


22-6 


27-0 


31-8 


36-8 


2 


5-5 


8-8 


12-3 


16-1 


20-3 


24-7 


29-5 


34-5 


39-9 


2i 


6-1 


9-7 


13-5 


17-6 


22-1 


26-8 


31-9 


37-3 


43-0 


2+ 


6-7 


10-6 


14-7 


19-2 


23-9 


28-9 


34-4 


40-0 


46-0 


2f 


7-4 


11-5 


16 


20-7 


25-7 


311 


36 8 


42-8 


491 


3 


8 


12-4 


17-2 


22'2 


27-6 


33-3 


39-3 


45-6 


52-2 


3i 


8-6 


12-3 


18-4 


23-8 


29-5 


35-4 


41-7 


48-3 


55-2 


n 


9-2 


14-2 


19-6 


25-3 


31-3 


37-6 


44-2 


51-1 


58*3 


n 


9-8 


15-2 


20-9 


26-9 


33-1 


39-7 


46-6 


53-8 


614 


4 


10-4 


16-1 


22-1 


28-4 


35-0 


41-9 


49-1 


56-6 


64-4 


4i 


11-1 


17-1 


23-4 


30-0 


36-9 


44-1 


51-6 


59-4 


67-6 


4^ 


11-7 


18-0 


24-5 


31-4 


38-7 


46-2 


54-0 


62-1 


70-6 


4f 


12-3 


18-9 


25-8 


33-0 


40-5 


48-3 


56-5 


64-9 


736 


5 


12-9 


19-8 


27-0 


34-5 


42-3 


50-5 


58-9 


67-6 


76-7 


H 


13-5 


20-7 


28-2 


36-1 


44-2 


52-6 


61-4 


70-4 


79-8 


H 


14-1 


21-6 


29-5 


37-6 


46-0 


54-8 


63-8 


73-2 


82-S 



Strength of Mater[als. 



149 





CAST-IRON PIPES. 






(Continued.) 




bore. 


i 
lbs. 


i 


i 


f 


1 


i 


1 


lbs. 


H 


in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


5i 


U'1 


22-6 


30-7 


39-1 


47-9 


66-9 


66-3 


76-0 


85-9 


6 


15-3 


23-5 


31-9 


40-7 


49-7 


591 


68-7 


78-7 


88-8 


H 


16-0 


244 


33-1 


422 


51-5 


61-2 


71-2 


81-2 


92-0 


6i 


16-6 


25-3 


34-4 


43-7 


53-4 


63-4 


73-4 


84-2 


95-1 


6i 


17-2 


26-2 


35 6 


45-3 


55-2 


65-3 


76-1 


87-0 


98-2 


1 


17-8 


27-2 


36-8 


46-8 


56-8 


67-7 


78-5 


89-7 


101-2 


n 


18-4 


28-1 


38-1 


48-1 


58-9 


69-8 


81-0 


92-5 


104-2 


n 


19-0 


29-0 


39-1 


49-9 


60-7 


72-0 


83-5 


95-3 


107-4 


n 


19-6 


29-7 


40-5 


51-4 


62-6 


74-1 


85-9 


98-0 


110-5 


8 


20-0 


30-8 


41-7 


52-9 


64-4 


76-2 


88-4 


100-8 


113-5 


8i 


20-9 


31-7 


43 


54-5 


66-3 


78-4 


90-8 


103-5 


116-6 


8* 


21-7 


32-9 


44-4 


56-2 


68-3 


80-8 


93-5 


106-5 


119-9 


8i 


22-1 


33-6 


45-4 


57-5 


700 


82-7 


95-7 


109-1 


122-7 


9 


22-7 


34-5 


46-6 


59-1 


71-8 


84-8 


98-2 


111-8 


125-8 


n 


23-3 


35-4 


47-9 


60-6 


73-6 


87-0 


100-6 


114-6 


128-9 


H 


23-9 


36-4 


49-1 


62-1 


75-5 


89 1 


103-1 


117-4 


131-9 


9f 


24-6 


37-3 


50-3 


63-7 


77-3 


91-3 


105-5 


120-1 


135-0 


10 


26-2 


38-2 


ol-o 


65-2 


79-2 


93-4 


108-0 


122-8 


138-1 


m 


25-8 


39-1 


52-8 


66-7 


81-0 


95-6 


110-4 


125-6 


141-1 


lOi 


26-4 


40-0 


540 


68-3 


82-8 


97-7 


112-9 


128-4 


144-2 


lOf 


27-0 


41-0 


55-2 


69-8 


84-7 


99-9 


115-4 


131-2 


147-3 


11 


27-6 


41-9 


56-5 


71-3 


86-5 


102-0 


117-8 


133-9 


150-3 


Hi 


28-2 


42-8 


57-7 


72-9 


88-4 


104-2 


120-3 


136-7 


153-4 


Hi 


28-8 


43-7 


58-9 


74-4 


90-2 


]06-3 


122-7 


139-4 


156-4 


m 


29-5 


44-6 


60-1 


75-9 


92-0 


108-5 


125-2 


142-2 


159-5 


12 


30-1 


45-6 


61-4 


77-5 


93-6 


110-6 


127-6 


145-0 


162-6 



f^treiigtlft of Jotiriials of Shafts. 

Mr. Buchanan's rule is: The cube i-oot of the weight in cwts. is 
nearly equal to the diameter of the journal; it being prudent to 
make the journal a little moi-e than less, and to make a due allow- 
ance for wearing. 

J5!r. What is the diameter of a journal of a water-wheel shaft, 13 
feet long, the weight of the wlieel being 15 tons!? 

By Mr. B.'s rule, 

y/15 X 20 = 6 7, or 7 inches diameter. 



13* 



150 Strength of Materials 



Weight in the middle, — — - x 13 rr: 873 £/873 — Hinches diam. 
oOO 



By Mr. Tredgold's rule, 

Qfif) 

— X 13 rr: 873^873.- 9^ 

3 / 

Weight equally distributed. 33600 x 13 r= 436800 —77- — = 
7-65 inches. ^^ 

To resist Torsion or Twisting. 

It is obvious that the strength of revolving shafts* is directly 
as the cubes of their diameters and revolutions ; and inversely as 
the resistance they have to overcome. 

Mr. Robertson Buchanan, in his Essay on the Strength of Shafts, 
gives the following data, deduced from several experiments, viz. : 
That the fly-wheel shaft of a 50-horse-power engine, at 50 revolu- 
tions per minute, requires to be 7^ inches diameter ; and therefore 
the cube of this diameter, which is = 421*875, serves as a multi- 
plier to all other shafts in the same proportion; and, taking this as 
a standard, he gives the following multipliers, viz. : 

For the shaft of a steam-engine, water-wheel, or any shaft connected with a first 

power, 400 

For shafts in inside of mills; to drive smaller machinerj', or connected with the 

shafts above, 200 

For the small shafts of a mill or machinery, 100 

From the foregoing, the following rule is derived, viz.: The 
number of horse power a shaft is equal to is directly as the cube of 
the diameter and number of revolutions ; and inversely as the 
above multipliers. 

Ex. 1. When the fly-wheel shaft of a 45-horse-power steam 
engine makes 90 revolutions per minute, what is the diameter of 
the journal? 



45 x 400 
90 ' 



r= 200 \/200 = Oy^Q inches diameter. 



Ex. 2. The velocity of a shaft is 80 revolutions per minute, and 
its diameter is 3 inches ; what is its power ? 

= 5*4 horse power. 

400 ^ 

Ex. 3. What will be the diameter of the shaft in the first 
example, when used as a shaft of the second mover, f 



5-8 , V45 x 200 .«.,.. 

= 4*64, or ~ rrr 4A inches diameter. 



1-25 ' 90 

* Shafts here are understood as the journals of shafts, the bodies of shafts being gene- 
rally made square. 

tThe diameters of the second movers will be found by dividing the numbers m the 
table by 125, and the diameters of the third movers, by dividing the numbers by 1 56. 



Diameters ok Shafts. 



151 



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rflH^c6«J"d5io * (^fflioH^?o~Jc?x>|lcoi-^«6o6(d 


o 
o 


OS OS- en C^ CJX ox >*«. ►;&► rf^ *«. 4^ CO 03 CO CO CO CO to to to 

to *obdsoxto«6a>>f^to 'Qods>fi^cotoh^c6-^dx 


o 

ox 



152 Strength of Wheals. 



It is a well known fact, that a cast-iron rod Avill sustain more 
torsional pressure than a malleable iron rod of the same dimen- 
sions ; that is, a malleable iron rod will be twisted by a less weight 
than what is required to wrench a cast-iron rod of the same 
dimensions. 

When the strength of malleable is less than that of cast iron to 
resist torsion, it is stronger than cast iron to resist lateral pressure, 
and that is in proportion as 9 is to 14. 

From the foregoing, it is easy for the millwright to make his 
shafts of the iron best suited to overcome the resistance to which 
they Avill be subject, and the proportion of the diameters of their 
journals, according to the iron of wliich they are made. 

Ex. What will be the diameter of a malleable iron journal to 
sustain an equal weight with a cast-iron journal of 7 inches dia- 
meter ? 

r' = 343. 

As 14 : 343 :: 9 : 220^ ; now y^220-5 = 6*04 inches diameter. 

Strength of Wheels. • 

The arms of wheels are as levers fixed at one end, and loaded at 
the other ; and, consequently, the greatest strain is upon the end of 
the arm next the axle. For that reason, all arms of wheels should 
be strongest at that part, and tapering toward the rim. 

The rule for the breadtli and thickness of arms, according to 
their length and number in the wheel, is as follows: Multiply the 
power or weight acting at the end of the arm by the cube of its 
length ; the product of which, divided by 2656 times the niunber of 
arms multiplied by the deflection, will give the breadth and cube 
of the depth. 

Ex. Suppose the force acting at the circumference of a spur- 
wheel to be 1600 lbs., the radius of wheel 6 feet, and number of 
arms 8, and let the deflection not exceed j-gth of an inch. 

-— — — = 163 =r breadtli and cube of the depth. 

2656 X 8 X 1 ^ 

Let the breadth be 2*5 inches; therefore = 65*2; which is 

2-0 

equal to the cube of the depth. Now the cube root of 65*2 is 
nearly 4*03 inches : this, consequently, is the depth or dimension 
of each arm in the direction of tiie force. 

Note.— When the depth at the rim is intended to be half that of the axes, use 1640 as a i 
divisor instead of 26.56. 

The teeth are as beams, or cantilevers, fixed at one end, and loaded at the other. The 
rule applying directly to them where the length of the beam is the length of the teeth 
and the depth the thickness of the teeth. For the belter explanation of the rule the 
(oilowin^i example is given. 

Ex. The greatest power acting at the pitch line of the wheel is 



Strength of "Wheels. 153 



6000 lbs., and the lhicknes«i of tlie teeth 1^ inch, tlie length of the 
teeth being 0*25 feet; it is required to determine tlie breadth of 
the teeth. 

6000x0-25 1500 „^. ^ ,, , 

rr rr 3'2 mches, the breadth required. 

212 X 1-5^ 477 ^ 

In order that the teeth may be capable of offering a sufficient 
resistance after being worn by friction, the breadth thus found 
should be doubled ; therefore, in the above example, the breadth 
should be 6*4, or say 6^ inches. 

The following data are gleaned fi'om experiments, which are 
therefore valuable, and of much use to the practical mechanic : 

Rule, Multiply the breadth of tlie teeth by the square of the 
thicliness, and divide the product by the length; the quotient will 
be the proportional strength in horse powei*, with a velocity of 
2*27 feet per second. 

&. What is the power of a wheel, the teeth of which are 6 
inches broad, IS inch thick, and 1'8 inch long, and revolving at 
the velocity of 3 feet per second? 

5^ X 6 13*5 

— — — X ■ =z 7*5, strength at 2*27 feet per second, then 

7*5 X 3 i 

2-27 : 7*5 :: 3 = -— = 9'91 horse power i 

2-27 ' 1 

Rule. The pitch is found by multiplying the thickness by 21, 
and the length is found by multi [flying the thickness b}' 1*2. 

£x. The thickness being 2 inches, what is the pitch and length ? 

2 X 2 1 =4-2, pitch. 

2 X 12 ■= 2-4, length. , 

For table of the proportions of wheels, see next page. 

-^^o^.— The breadth of the teeth, as commonly executed by the best mechanic?, 
seems to be from about twice to thrice the pitch. 



Bean Shot Copper — Take copper, melt it, and pour it in a small 
stream into boiling water. 

Feather Shot Copper. — Take copper, melt it, and pour it in a 
small stream into cold water. 

To preserve Walls from Dampness. — When the walls are about 
two feet high, use for one row of stones or bricks a mixture of tar. 
pitch, and fine sand, in the same way as mortar. The composition 
must be previously melted to a pro})er consistence. 

To prevent Iron kuom Rusting. — Warm your iron till you 
cannot bear your hand on it wilhuut burning yourself. Then rub 
it with new and clean white wax. Put it again to tlie fire till it 
has soaked in the wax. When done rub it over with a piece of 
berge. This prevents the iron from rusting afterwards. 



154 



Alloys. 
Table of the Proportions of Wheels. 






T 




^ 


4-2 


2- 


8- 


3 99 


1-9 


7-6 


3-78 


1-8 


7-2 


3 57 


1^ 


68 


3-36 


16 


64 


315 


1-5 


6- 


2-94 


1-4 


56 


' 2 73 


1-3 


5 2 


2-52 
2-31 


1-2 
11 


ti 


210 


1- 
•9 
•8 


IV' 

3^ 


1-47 


•7 


2-8 


^' 1-05 


'6 
'5 


1^ 






2-40 


13-33 


17-61 


35-23 


2-28 


13-03 


15-90 


31 -SO 


2li 


1080 


14 27 


28-54 


2-04 


9-63 


12-72 


25-54 


1^2 


8 53 


1127 


22-54 


1-80 , i 


. 7-50 


9-91 


19-82 


1-68 


: 6-53 


8 63 


17-26 


1-56 


5-63 


7-44 


1488 


1-44 


4-80 


6-34 


12-68 


1-3^ 


4 03 


5-32 


10 64 


ii; 


. 3 33 


4-40 
3 57 


8-81 
7-14 


^« J 


213 


2*81 


5-62 


•84 '^ 


1-63 


215 


4-30 


•sr2 i 


1-20 


1-59 


318 


•60 '^ 


•83 


110 


2-20 



ALLOYS, OE MISCELLAXEOUS METALS. | 

ChaudeCs Medal Metal 
Copper 100 parts; tin 4-17. Cast in moulds formed of eopel 
bone ash. , 

Jj^ad in Grains. ^ 

Lead, melt it, and ponr it in a small stream from a height of 
three or four feet into cold water. 

Bell Metals. 

1. Copper 25 parts; tin 5. Mix. 

2. Copper 79 parts; tin 26. Mix. 

3. Copper 78 parts : tin 22. Mix. 

Coimnon Bell Metal. 
Copper 100 parts ; tin 50. Mix. 

Parisian Bell Metal. 
Copper 72 parts; tin 26^; iron \l. This alloy is used for the 
bi^Ds of small ornamental clocks. 

Bath Metal. 
Bi-ass 32 parts ; spelter 9. Mix. ^ 



Alloys, 



155 



Brass 86 parts ; zinc 9. 
Copper 3 parts. 



Another, 

Mix. 

Brass. 

Melt, then add zinc 1 part 

Button Makers^ Fine Brass, 

Brass 8 parts; zinc 5. Mix. 

Button Maker i< Common Brass* 
Button brass 6 parts; tin 1 ; lead 1. Mix. 

Bright Brass Color. 
Brass reduced to fine powder. 

Bed Brass Color* 
Copper filings 3 parts ; bole 2. Mix. 
Fine Brass. 
Copper 2 parts ; zinc 1. Mix. 

Brass for Wire. 
Copper 34 parts ; calamine 56. Mix. 

To give Plates of Copper a Brass Color. 
Expose the plates, after being sufficiently heated, to the fumes of 
zinc. 

To Brass Copper Vessels. 
Argol 1 part; amalgam of zinc 1 ; muriatic acid 2 ; water to fill 
the vessel. Mix. 

Brass or Hard Solder. 
Brass 2 parts ; zinc 1. A little tin is occasionally added. 

Jewellers' 3fetal. 
Copper 30 parts; brass 10; tin Y. Mix. 
Fusible Alloys. 

1. Bismuth 8 parts; lead 5; tin 3. This is fusible at boiling 
water heat. 

2. Zinc, lead, and bismuth equal parts. This may be fused in a 
bit of writing paper, and will melt even in hot water. 

3. Lead 3 parts; tin 2; bismuth 5. Mix. This alloy melts at 
197° Fall. In using this composition to make casts of seals, gems, 
<fec., it should be employed at the lowest possible temperature at 
which it will keep fluid ; for this purpose it is as well to let it 
become pasty, and then forcibly impress the substances together. 

4. Bismuth 2 parts ; tin 3 parts ; lead 5. Melt. This alloy fuses 
in boiling water. 

German Silver. 

1. Nickel 1 part ; zinc 1 ; copper 2. 

When intended for rolling into plates, use the following: 

2. Nickel 25 parts ; zinc 20 ; copper 60 ; to which may be added 
3 of lead. 

3. Pure copper 55 parts ; nickle 23 ; zinc 17 ; iron 3 ; tin 2. 

Fine White German Silver. 
Iron 1 part; nickel 10; zinc 10; copper 20. Mix. 



156 Alloys. 

I German Silver for CaiihvgSj <Scc. j 

j Lead 3 parts ; nickel 20 • zinc 20 ; copper 60. Mix. j 

I Gemiine German Silver. I 

Copper 40^ parts : nickel 81^ • zinc 25^ ; iron 2|. Mix. 

Gilding Metal. \ 

Copper 4 parts; brass 1 : tin 1. Fuse together. I 

Another. \ 
Copper 14 parts ; zine 6 ; tin 4. 

To Separate Gold from Gilt Copper or Silver. 
Take a solution of borax in water, apply to the gilt surtace, and 
sprinkle over it some finely powdered sulphur; make the article 
i red hot, and quench it in water ; then scrape off the gold, and 
I recover it by means of lead. 

I Gold in Grains. 

Gold 3 parts; silver 1. Granulate by pouring it in a small 
i stream, from a moderate height, into cold water; then dissolve the 
I silver with nitric acid, and wash well in pure water; next heat the 
' grains, to give them a proper lustre. 

Common Gold. 
Spanish copper 16 parts: silver 1 : gold 2. Melt together. 

Onian'a Fusible Metal. 
Tin 2 parts; lead 3 ; bismuth 5. Melt. This alloy melts at 197° 
Fall. The addition of a little mercur}' renders it still more fusible. 

Alloy for Flute Key Valves. 
Lead 4 parts ; antimony 2. Fuse. 
Pewter. 

1. Tin 100 parts; antimony 17. Mix. 

2. Zinc 1 part; copper 3 ; lead 8 ; tin 60. Melt the copper, then 
add the rest. 

3. Fhie. Tin 50 parts, antimony 4 : bismuth 1 ; copper 1. Mix, 
as before. 

4. French. Lead 9 parts; tin 41. Mix. 

Kellers Medal Alloy. 
Tin 9 parts ; copper 89 ; zinc 2. 

Gun Metal. 
Brass 100 parts; spelter 13 ; tin 6. Mix. 

Another. 
Copper 9 parts; tin 1. 

Pinchbeck. 

1. Brass 2 parts; copper 3. Melt under charcoal dust 

2. Copper 5 parts: zinc 1 Melt the copper, then add the zinc 

Tin Filings. 
Take grain tin. melt it in an iron vessel, and stir it, while 
cooling, until it becomes a powder ; then sift it 



Alloys. loT 



Ihi in Grains. 
Take Cornish grain tin, melt it, and pour it into a wooden box, 
well rubbed on the inside with whiting or chalk ; close the cover, 
and continue shaking it violently until the tin is reduced to 
powder ; then wash it in clean water, and dry it immediately. 
Mosaic Gold, or Moln. 
Take copper and zinc, equal parts. Melt at the lowest tempera- 
ture that will fuse the former ; then mix by stirring, and add more 
zinc, until the fused alloy becomes perfectly white ; lastly, pour it 
into moulds. The proportion of zinc to the copper is from 60 to 55 
per cent., exclusive of what is lost by the heat employed. 
Hard White Metal, 
Tin 1 part ; spelter 3 ; brass 20. Mix. 
Turners Brass. 
Brass 98 parts ; lead 2. Mix. 

Tltania, or Britannia Metal, 

1. Plate brass 2 parts ; tin 2 ; bismuth 2 ; antimony 2 ; copper 
1 ; arsenic 1. Mix, and add this alloy, at discretion, to melted tin. 

2. Spanish. Of Spanish Titania metal there are two kinds. The 
first is made thus : Antimony 4 parts ; tin 2 ; arsenic 1. The 
second is made in the following manner: Scrap iron 1 part; 
antimony 2; nitre a little. Melt, and harden one pound of tin 
with 2 oz. of this composition. A little arsenic improves the color 
of this alloy. 

Tutenag. 
Tin 2 parts; bismuth!. Fuse. 

Type Metal. 
Lead 11 parts; antimony 2. Fuse. 

Ring Gold. 
Spanish copper 6 parts ; silver 3 ; gold 5 ; Mix, 

Priyice Ruperfs Metal. 
Copper 2 parts ; melt, and add zinc 1 part. 

White Metal. 
Brass 1 part ; tin 2 ; antimony 4. 

Another. 
Lead 20 parts ; bismuth 12 ; antimony 1. Fuse. 

Yellow Dipping Metal. 
Copper 19 parts; spelter 6. Mix. 

A Metal that resetnbles Silver. 
Tin f oz. ; copper 1 lb. This alloy will make a pale bell metal 
that will roll and ring very near to sterling silver. 

Silver Dust. 
Take silver, dissolve it in nitric acid, and precipitate it with 
slips of bright copper ; wash the powder in spirits, and dry it. 
Imitation Platina, 
Pale brass 8 parts ; spelter 5. Mix. 

14 ' 



158 Alloys. 

Dessaiissi/s Steel. 
Copper 100 parts; tin 14. This alloy may be hardened and 
sharpened in a similar way to steel. 

Stereotupe Metal. 
Lead 18 parts; antimony 4 parts; bismuth 2 parts. Melt. 

Another. 
Lead 16 parts; antimony 3 parts; tin 5 parts ; copper 2 parts. 

Anotlier. 
Lead 20 parts; tin 8; antimony 1. 

Speculum Metal. 
Copper 43 parts ; tin 20. IAia. 

Another. 
Copper Y parts; melt, and add zinc 3 parts, tin 4. 

Prince's Metal. 
Copper 3 parts ; zinc 1. 

Another. 
Brass 8 parts; zinc 1. 

Another. 
Zinc and copper, equal parts. Mix. 

To make Iron resemble Gold. 
Take of linseed oil 3 oz. ; tartar 2 oz. ; yolk of eggs, boiled hard 
and beaten, 2 oz. ; aloes |- oz. ; saffron 5 grains ; turmenc 2 grains. 
Boil together in an earthen vessel, and with it wash the iron, and 
it will look like gold. Should there not be linseed oil enough more 
may be added. 

Queen's Metal. 
Lead 1 part ; bismuth 1 ; antimony 1 ; tin 9. Mix. 

Ar^other. 
Tin 9 parts; bismuth 1 ; lead 2 ; antimony 1. Mix by melting. 

Another. 
Tin 1000 parts; regulus of antimony 80; bismuth 10; copper 
40. Melt the copper, then expertly add the rest, and mix Well 
together. 

Purified Quicksilver. 
Quicksilver 1 part ; iron filings 1. Distil in an iron retort, into 
a vessel containing water. 

Mock Gold. 
Platina 7 parts ; copper 16 ; zinc 1. Fuse together. 

Bronze Metals. 
For medals, and small castings — copper 95 parts ; tin 4, 

Another. 
Copper 89 parts ; tin 8 ; zinc 3. 

Another. 
Ancient Copper 100 parts; tin 7 ; lead 7. 



Alloys. 159 

Another. 
Kelly s. Copper 91 parts; zinc 6 ; tin 2 ; lead 1. 

Blanched Copper. 
Copper 8 parts ; arsenic \ part. 

Manheini Gold. 
Copper 3 parts ; zinc 1. Melt separately, then suddenly mix 
them, and stir well. 

Red Tomhac. 
Copper 11 parts; zinc 2. Mix. 



Furniture Paste. 

1. Melt 1 pound of beeswax with ^ pint of linseed oil, and add -J 
oz. alkanet root ; keep it at a moderate heat till sufficiently 
coloied , then remove from the fii-e, add ^ pint of oil of turpentine, 
strain through muslin; and put it into small gallipots to cool.^ 

2. Sciape 4 oz. of wax, and put it into a pipkin with as much 
oil of turpentine as will cover it, and ^ oz. of powdered resin ; melt 
with a gentle heat, and stir in sufficient Indian red to color it. 

3. Equal weights of beeswax, spirit of turpentine, and linseed 
oil. 

Bronze Powder. 

The best methods of preparing these powders are probably kept 
secret. The following are some of the published recipes : 

1. Gold leaf, oi* alloys of gold, reduced to powder by grinding 
them with sulphate of potash, or with honey, and washing away 
the extraneous matter with hot Avater, and drying the metallic 
powder. 

2. Dutch metal, and other similar alloys, treated in the same 
way. 

S. Verdigris 4 oz. ; tutty 2 oz. ; sublimate 1 dr. ; borax 1 dr. ; 
nitre 1 dr. Mix them into a paste with oil, and fuse the mixture in 
a crucible. This has failed in some hands, perhaps from the tutty 
being factitious. 

4. Mix together 100 parts of sulphate of copper, and 50 of 
cr^'stallized carbonate of soda ; apply heat till they unite. Powder 
the mass, when cold, and add 15 })arts of copper hlings ; mix Avell, 
and keep it at a white heat for twenty minutes. Wash and dry 
the product. 

Balls for Scouring — Breeches Balls, Clothes Balls. 

1. Bathbrick 4 i>art8 ; pipeclay 8 ; pumice 1 ; softsoap 1 ; ochre, 
umber, or other color, to bring it to the desired shade, q. s. ; ox-gall 
to form a paste. Make it into balls, and dry them. 

2. Pipeclay 4 oz. ; fuller's-earth ^ oz. ; whiting I oz. ; white 
pepper -| oz ; ox-gall sufficient to foiin it into a paste. 

3. Pipeclay 3 oz. ; white pepper 1 dr. ; starch 1 dr, onis powder 
\\ dr. It may be kept in powder, or formed into balls, ns above. 



160 



Mensuration of Circles. 







MENSURATION OF CIRCLES. 






Table of the Diameters, Circumferences, an 


d Areas of Circles. \ 


I n 


ij 

t - C 


III 


it 


'5 - 


ill 


11 


.1= B 


•SgS 


1 






4 


12-566 


12-566 


9 


28-274 


63-617 


! h 


•1963 


•00306 


i 


12-959 


13 364 


i 


28-667 


65-396 


i 

1 t) 


•392Y 
•5890 


•01227 
•02761 


i 


13-351 


14-186 
15-033 


JL 

4 

1 


29059 
29-452 


67-200 
69-029 


f 


13-744 


i 


•YSoi 


•04909 


i 


14137 


15-904 


i 


29-845 


70-882 


A 


•9817 


•07670 


1 


14-529 


16-800 


i 


30-237 


72-759 


t 


1-1781 


•11044 


■ f 


14-922 


17-720 


f 


80 630 


74-662 


7 

re 


1-8744 


•15033 


: i 


15-315 


18-665 


i 


31-023 


76-588 


J 


1-5708 


•19635 


5 


15 708 


19-635 


10 


31-416 


78-540 


A 


1-7671 


•24850 


i 


16-100 


20-629 


i 


31-808 


80 515 


t 


1-9635 


-30680 


i 


16-493 


21-647 


i 


32-201 


82 516 


1 1 

1 6 


2 1598 


•37122 


f 


16-886 


22-690 


1 


33-594 


84-540 


f 


2-3562 


-44172 


i 


17-278 


23-758 


i 


32986 


86-590 


13 


2-5525 


•51849 


1 


17-671 


24-850 


i 


33-379 


88-664 


A 


2-7489 


•60132 


f 


18 064 


25-967 


i 


33-772 


90-762 


!-l 


2-9452 


•69030 


! i 


18-457 


27-108 


i 


34-164 


92-885 


1 


3-141 


•785 


6 


18-849 


28-274 


11 


34-657 


95-033 


i 


3-534 


•994 


i 


19-242 


29-464 


i 


34-950 


97-205 


i 


3-927 


1-227 


i i 


19-635 


30-679 


i 


35-343 


99-402 


f 


4-319 


1-484 


i f 


20 027 


31-919 


1 


35-785 


101 623 


1 


4-712 


1-767 


i 


20-420 


33-188 


i 


36-128 


103-869 


f 


5-105 


2-073 


f. 


20-813 


34-471! 


•1 


36-521 


106-139 


f 


5-497 


2-405 


: f 


21205 


35-784; 


1 


36-913 


108-434 


1 


5-890 


2-761 


1 i 


21-598 


37-122 


1 


37-306 


110-763 


2 


6-283 


3141 


: 7 


21-991 


38 484 


12 


37-699 


113-097 


i 


6-675 


3-546 


: i 


22-383 


39-871 


i 


38-091 


115-466 


i 


7-068 


3-976 


i 


22-776 


41-282 


J 


38-484 


117-859 


f 


7-461 


4-430 


f 


23-169 


42-718 


f 


38-877 


120-276 


i 


7-854 


4-908 


i i 


23 562 


44-178 


i 


39-270 


122718 


f 


8-246 


5-411 


^ i 


23-954 


45-663 


f 


89 662 


125-184 


f 


8-639 


5-989 


: f 


24 347 


47-173 


f 


40-055 


127-676 


^ 


9-032 


6 491 


1 * 


24-740 


48-707 


1 


40 448 


130-192 


3 


9-424 


7-068 


1 8 


25-132 


50-265 


13 


40-840 


132-732 


i 


9-817 


7-669 


i i 


25-525 


51-848 


i 


41-233 


135^297 


i 


10-210 


8-295 


; i 


25-918 


53-456 


i 


41 626 


137^886 


f 


10-602 


8-946 


i -1 


26-310 


55-088 


f 


42018 


140-600 


i 


10-995 


9-621 


1 i 


26-703 


56-745 


i 


42-411 


143-139 


f 


11-388 


10-320 


f 


27-096 


58-4261 


f 


42-804 


145-802 


i 


11-781 


11-044 


f 


27-489 


60-132' 


f 


43-197 


148-489 


i 


12^173 


11-793 


i 


27-881 


61-862! 


i 


43-589 


151-201 



Mensuration of Cirolks. 



161 



i 


















Piam. 


Gircura. 


Area. 


Diam. 
19 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


u 


43-98 


153-93 


59-69 


283-52 


24 


75-39 


452-39 


i 


44-:i7 


156-69 


i 


60-08 


287-27 


i 


75-79 


457-11 


i 


44-76 


159-4S 


i 


60-47 


291-03 


i 


76-18 


461-86 


f 


45-16 


162-29 


1 


60-86 


294-83 


f 


76-57 


466-6^3 


i 


40-55 


165-13 


i 


61-26 


298-64 


i 


76-96 


471-43 1 


f 


45-94 


167-98 


t 


61-65 


302-48 


i 


77-36 


476-25 


* 


46-83 


17087 


f 


62-04 


306-35 


f 


77-75 


481-10 


i 


46-73 


173-78 


i 


62-43 


310-24 


i 


78-14 


485-97 


15 


47-12 


176-71 


20 


62-83 


314-16 


25 


78-53 


490^87 j 


* 


47-51 


179-67 


i 


63-22 


318-09 


i 


78-93 


495-79 


i 


47-90 


182-65 


i 


63-61 


322-06 


i 


79-32 


500-74 


f 


48 30 


185 66 


f 


6401 


32605 


i 


79-71 


505-71 


* 


48-69 


188-69 


i 


64-40 


330-06 


i 


8010 


510-70 j 


* 


49-08 


191-74 


f 


64-79 


334-10 


f 


80-50 


515-72 


f 


49-48 


194-82 


f 


65-18 


338-16 


f 


80-89 


520-76 


i 


49-87 


197-93 


i 


65-58 


342-25 


1 


81-28 


525-83 


16 


50-26 


201-06 


21 


65-97 


346-36 


26 


81-68 


530-93 


i 


50-65 


204-21 


i 


66-36 


350-49 


i 


82-07 


536-04 ! 


i 


51-05 


207-39 


4 


66-75 


354-65 


i 


82-46 


541-18 i 


f 


51-44 


210 59 


f 


67-15 


358-84 


» * 


82-85 


546-35 ■ 


i 


51-83 


213-82 


* 


67-54 


363 05 


* i 


83-25 


551-54 .' 


f 


5222 


217-07 


f 


67-93 


367-28 1 


i 


83-64 


556-76 ; 


f 


52-62 


220-35 


f 


68-32 


371 54 


f 


84-03 


562-00 i 


i 


53-01 


223-65 


i 


68-72 


375-82 


i 


84-43 


567-26 


17 


58-40 


226-98 


22 


69-11 


38013 


27 


84-82 


572-65 


i 


53 79 


230-33 


i 


69-50 


381-46 


i 


85-21 


577-87 


i 


54-19 


233-70 


i 


69-90 


388-82 


i 


85-60 


583-20 


f 


54-58 


237-10 


f 


70-29 


393-20 


f 


86-00 


588-57 


i 


54-97 


240-52 


i 


70-68 


397-60 


i 


86.39 


593 95 


f 


55-37 


243-97 


i 


71-07 


402-03 


i 


86-78 


599-37 


f 


55-76 


247-45 


f 


71-47 


406-49 


f 


87-17 


604-80 


* 


56-15 


250-94 


1 


71-86 


410-97 


i 


87-57 


610-26 


18 


56-54 


254-46 


23 


72-25 


415-47 


28 


87-96 


615-75 


i 


5(V94 


258-01 


i 


72-64 


420-00 


i 


88 35 


621-26 


i 


57 33 


261-58 


i 


73-04 


424-55 


i 


88-75 


626-79 


f 


57-72 


265-18 


f 


73 43 


429-13 


i 


89-14 


632-35 


i 


58-11 


268 80 


i 


73-82 


433-73 


i 


89-53 


637-94 


f 


58-51 


272-44 


1 


74-21 


438-36 


1 


89-92 


643-54 


4 


58 90 


276-11 


i 


74 61 


443-01 


f 


90 32 


649-18 


* 


59-29 


279-81 


i 


75-00 


447 69 


i 


90-71 


654-83 


1 



14* 



'm 



Mensuration of CiROtES. 





Diam. 


Circum. 


Area. 


1 
Diam. 


circum. 


Area, 


Diam.j C 


ircum. 


Area. 


29 


91-10 


660-52 


34 


106-8 


907-92 


39 


122-6 


1194-59 


i 


91-49 


666-22 


i 


107-2 


914-61 


i 


122-9 


1202-26 


i 


91-89 


671-95 


i 


107-5 


921-32 


i 


123-3 


1209-95 


f 


92-28 


677-71 


« 


107-9 


928-06 


f 


123-7 


1217-67 


i 


92-67 


683-49 


i 


108-3 


9b4-82 


i 


124-0 


1225-42 


f 


93-06 


689-29 


i 


108-7 


941 60 


i 


124-4 


1233-18 


f 


93-46 


695-12 


f 


1091 


948-41 


f 


124-8 


1240-98 


i 


93-85 


700-98 


i 


109-6 


965 25 


i 


125-2 


1248-79 


30 


94-24 


706-80 


35 


109-9 


962-11 


40 


125-6 


1256-64 


i 


94-64 


712-70 


i 


110-3 


968-99 


i 


126-0 


1264-50 


i 


95-03 


718-60 


i 


110-7 


975-90 


i 


126-4 


1272-39 


1 


95-42 


724-64 


f 


111-1 


992-84 


1 


126-8 


1280-31 


i 


95-81 


730-61 


i 


1115 


989-80 


i 


127-2 


1288-25 


f 


96-21 


736-61 


f 


111-9 


996-78 


f 


127-6 


1296-21 


f 


96-60 


742-64 


f 


112-3 


1003-71 


f 


128-0 


1304-20 


i 


96-99 


748-69 


i 


112-7 


1010-81 


i 


128-4 


1312-21 


31 


97-38 


754-76 


36 


113-0 


101787 


41 


128-8 


1320-26 


i 


97-78 


760-86 


i 


113-4 


1024-95 


i 


129-1 


1328-32 


i 


98-17 


766-99 


i 


113-8 


1032-06 


i 


129-5 


1336-40 


f 


98-56 


773-li 


1 


114-2 


1039-19 


1 


129-9 


1344-51 


i 


98-96 


779-31 


i 


114-6 


1046-39 


i 


130-3 


1352-65 


f 


99-35 


785-51 


i 


115-0 


1053-62 


f 


130 7 


1360-81 


f 


99-74 


791-73 


f 


115-4 


1060-78 


f 


131-1 


1369-00 


1 


100-13 


797-97 


i 


. 115-8 


1067-95 


i 


131-5 


1377-21 


32 


100-5 


804-24 


37 


116-2 


1076-21 


42 


131-9 


1385-441 


i 


100-9 


810-54 


i 


116-6 


1082-48 


i 


132-3 


1393-701 


J 


101-3 


816-86 


i i 


117-0 


1089-79 


i 


132-7 


1401981 


•1 


101-7 


823-21 


\ i 


117-4 


1097-11 


f 


133-1 


1410-29 


i 


102-1 


829-57 


i 


117-8 


1104-46 


i 


133-5 


1418-62 


f 


102-4 


835-97 


f 


118-2 


1111-84 


i 


1339 


1426-98 


f 


102-8 


842-39 


f 


118-5 


1119-24 


f 


134-3 


1436-361 


1 


103-2 


848-83 


i 


118-9 


1126-66 


i 


134-6 


1443-77 \ 


33 


103-6 


865-30 


38 


119-3 


1134-11 


43 


135-0 


1452-20 


i 


104-0 


861-79 


i 


1197 


1141-59 


i 


136-4 


1460-65 


i 


104-4 


868-30 


i 


120-1 


1149-08 


i 


136-8 


1469-13 


1 


104-8 


874-84 


f 


120-6 


1166-61 


f 


1362 


1477-63 


i 


105-2 


881-41 


i 


120-9 


1164-16 


i 


136-6 


1486-17 i 


f 


105-6 


888-00 




121-3 


1171-73 


i 


137-0 


1494-72 i 


f 


106-0 


894-61 


J 


121-7 


1179-32 


f 


137-4 


1603-30 


i 


106-4 


901-25 


1 


1221 


1186-94 


i 


137-8 


1511-90 

i 


1 



Mensuration Of Circles. 



163 



Diam. 



Circum. 



44 

i 
i 
I 
i 

f 

i 

45 

i 
i 
f 
i 



138-2 
138-6 
139-0 
139 '4 
139-8 
140-1 
140-5 
140-9 

141-3 
1417 
142-1 
142-5 
142-9 
143-3 
143-7 
1441 



Area. 


Diam. 
46 


Circum. 


Area. 


Diam. 


Circum. 


1520-53 


144-5 


1661-90 


48 


150-7 


1529-18 


i 


144 9 


1670 95 


i 


151-1 


1537-86 


J 


145-2 


1680-01 


i 


151-5 


1546-55 


1 


145-6 


168910 


i 


151-9 


1555-28 


^ 


146-0 


1698-23 


i 


152-3 


156403 


1- 


146-4 


1707-37 


i 


152-7 


1572-81 


f 


146-8 


1716-54 


i 


153-1 


1581-61 


i 


147-2 


1725 73 


i 


153-5 


1590-43 


47 


147-6 


1734-94 


49 


153-9 


1599-28 


i 


148-0 


1744-18 


i 


154-3 


1608-15 


i 


148-4 


1753-45 


i 


154-7 


161704 


f 


148-8 


1762-73 


t 


155-1 


1625-97 


-^ 


149-2 


1772-05 


i 


155-5 


1634-92 


f 


149-6 


1781-39 


* 


155-9 


1643-89 


1 


150-0 


1790-76 


i 


156-2 


1652-88 


1 


150-4 


1800-14 


i 


156-6 



Area, 

1809-56 I 
1818-99 : 
1828-46 i 
1837-93 ! 
1847-45 i 
1856-99 
1866-55 
1876-13 

1885-74 
1895-37 
1905-03 
1914-70 
1924-42 
1934-15 
1943-91 
1953-69 



Diam. 



60 
i 
i 
f 

51 
i 
i 
f 

52 

i 



53 

i 



54 
i 



Circum. 
inches. 



157-0 
157-8 
158-6 
159-4 

160-2 
161-0 
161-7 
162*5 

163-3 
164-1 
164-9 
165-7 

166-5 
167-2 
168-0 
168-8 

169-6 
170-4 
171-2 
172-0 



Area in 
square in. 

1963-5 
3983-1 
2002-9 
2022-8 

2042-8 
2062-9 
2083-0 
2103-3 

2123-7 
2144-1 
2164-7 
2185-4 

2206-1 
2227-0 
2248-0 
2269 

2290-2 
2311-4 
2332-8 
2354-2 




14-18 
14-32 
14-46 
14 60 

14-74 
14 89 
15-03 
15-17 

15-32 
15-46 
15-61 
15-75 

15-90 
16-05 
16-20 
16-34 



Circum. 
inches. 



172-7 
173-5 
174-3 
175-1 



Area in 
square in. 

2375-8 
2397-4 
2419-2 
2441-0 



175-9 


2463-0 


176 7 


2485-0 


177-5 


2507-1 


178-2 


2529-4 


179-0 


2551-7 


179-8 


2574-1 


180-6 


2596-7 


181-4 


2619-3 


182-2 


2642-0 


182-9 


2664-9 


183-7 


2687-8 


184-5 


2710-8 


185-3 


2733-9 


186-1 


2757-1 


186-9 


2780-6 


187-7 


2803-9 



16-49 
1664 
16-80 
16-95 

17-10 
17-25 
17-41 
17-56 

17-72 

17-87 
1803 
18-19 

18-34 

18 50 

18-68 

1882 

18-98 
19-14 
19-30 

19 47 



164 



Mexsuration of Circles. 





Diam 


Circum. 


Area in 


Area in 


Diam 


Circum. 


Area in 


Area m 


in. 


inches. 


j square in. 


square feet. 


in. 

69 


inches. 


square in. 


square feet. 


60 


188-4 


2827-4 


19-63 


216-7 


3739-2 


25-96 


i 


189-2 


2851-0 


19-79 


i 


217-5 


3766-4 


26-16 


i 


190-0 


2874-7 


19-96 


i 


218-3 


3793-6 


26-34 


i 


190 8 


2898-5 


20-12 


f 


219-1 


38210 


26-53 


61 


191-6 


2922-4 


20-29 


70 


219-9 


3848-4 


26-72 


i 


192-4 


2946-4 


20-46 


i 


220-6 


3875-9 


26-91 


i 


193-2 


2970-5 


20-62- 


i 


221-4 


3903 6 


27-10 


i 


193-9 


2994-7 


20-79 


f 


222-2 


3931-3 


27-30 


62 


194Y 


30190 


20-96 


71 


223-0 


3959-2 


27-49 


i 


195-5 


3043-4 


21-13 


1 i 


223-8 


3987-1 


27-68 


i 


196-3 


3067-9 


21-20 


i i 


224-6 


4015-1 


27-87 


i 


197-1 


3092-5 


21-47 


i ^ 


225-4 


4043-2 


28-07 


' 63 


197-9 


3117-2 


21-64 


\ 72 


226-1 


4071-5 


28-27 


■ i 


198-7 


3142-0 


21-81 


1 i 


226-9 


4099-8 


28-47 


I 


199-4 


3166 9 


21-98 


i i 


227-7 


4128 2 


28-66 


^ 1 
64 


200-2 
201-0 


3191-9 
3216-9 


22-16 
22-34 


1 ^ 
73 


228-5 
229-3 


4156-7 


28-86 


4185-3 


29-06 


i 


201-8 


3242-1 


22-51 


i 


230-1 


4214-1 


29-26 


i 


202-6 


3267-4 


22-68 


i 


280-9 


4242-9 


26-46 


i 


203-4 


3292-8 


22-86 


1 


231-6 


4271-8 


29-66 


65 


204-2 


3318-3 


23-04 


74 


232-4 


4300-8 


29-86 


i 


204-9 


3343-8 


23-22 


i 


233-2 


4329-9 


30-06 


i 


205-7 


3369-5 


23-39 


i 


234-0 


4359-1 


30-26 


f 


206-5 


3395-3 


23-57 


f 


234-8 


4388-4 


30-47 


66 


207-3 


3421-2 


23-75 


75 


235-6 


4417-8 


30-67 


i 


208-1 


3447-1 


23-93 


i 


236-4 


4447-3 


30-88 


i 1 


208-9 


3473-2 


24-11 


i 


237-1 


4476-9 


31-09 


f 


209-7 


3499-3 


24-30 


f 


237-9 


4506-6 


31-30 


67 


2104 


3525-6 


24 48 


76 


238-7 


4536-4 


31-50 


i 


211-2 


3552-0 


24-66 


i 


239-5 


4566-3 


31-71 


i 


2120 


3578-4 


24-84 


i 


240-3 


4596-3 


31-91 


1 1 

1 i 


212-8 


3605-0 


25-03 


f 


241-1 


4626-4 


3212 


68 ! 


213-6 


3631-6 


25 22 ! 


77 


241-9 


4656-6 


32-33 


i 


214-4 


3658-4 


25-40 


i 


242-6 


4686 9 


3-2-54 


i 


215-1 


3685-2 


25-59 


\ 


243-4 


4717-2 


32-75 


1 


2159 


3712-2 


25-77 

1 


* 


244-2 


4747-7 


32-96 



Mensuration of Circles. 



1C5 



















Diam 


Circum. 


Area in 


Area in 


Diann 


Circum 


Area in 


i Area in 


in. 


inches. 


squjire in. 


square lee 


in. 


incl^ieR. 


square in 


. I square feet. 


78 


245 


4778-3 


33-18 


' 87 


278-3 


5944-6 


41-28 


i 


245-8 


4809-0 


38-39 


i 


2741 


5978-9 


41 52 


i 


246-6 


4839-8 


38 •60 


i 


274-8 


6013-2 


i 41-75 


1 


247-4 


4870-7 


33-81 


f 


275-6 


6047-6 


41-99 _ 


79 


248-1 


4901-6 


34 08 


I 88 


276-4 


6082-1 


1 42-28 


i 


248-9 


4932-7 


34-24 


i 


277-2 


6116-7 


! 42-47 


i 


249-7 


4963-9 


34 46 


\ 


278-0 


6151-4 


1 42-71 


i 


250-5 


4995-1 


84-68 


1 -^ 


278-8 


6186-2 


42 95 


80 


251-8 


5026-5 


34-90 


89 


279-6 


6221-1 


48-2^ 


i 


252-1 


5058-0 


8512 


i i 


280-3 


6256-1 


1 48-44 


i 


252-8 


5089-5 


85-84 


i * 


281-1 


6291-2 


1 48-68 


f 


258-6 


5121-2 


85-56 


1 


281-9 


6326-4 


48-92 


81 


25 


4-4 


5153-0 


85-78 


90 


282-7 


6361-7 


1 44 17 


i 


25 


5-2 


5184-8 


36-00 


i 


28 


8-5 


6897^1 


44-42 


i 


25 


6-0 


5216-8 


86-22 


\ 


1 28 


4-3 


6432-6 


, 4466 


i 


25 


6-8 


5248-8 


36 44 


f 


28 


5-1 


6468-2 


44-81 


82 


25 


7-6 


5281-0 


36-67 


91 


28 


5-8 


6503-8 


45-16 


i 


25 


8-3 


5818-2 


86 90 


4 


286-6 


6539-6 


1 45-41 


i 


25 


9-1 


5345-6 


87 12 


i 


287-4 


6575-5 


45-66 


i 


25 


9-9 


5378-0 


87-84 


I 


288-2 


6611-5 


45-91 


83 


26 


0-7 


5410-6 


37-57 


92 


289-0 


6647-6 


46-16 


i 


261-5 


5443-2 


87-79 


I 


289-8 


6683-8 


46 41 i 


i 


262-8 


5476-0 


88-02 


\ 


290-5 


6720-0 


46-66 


i 


263-1 


5508 8 


88-25 


f 


291-3 


6756-4 


46-91 


84 


263-8 


55417 


38-48 


93 


292-1 


6792-9 


47-17- 1 


i- 


2646 


5574-8 


38-71 


i 


292-9 


6829-4 


47-43 


i 


265-4 


5607-9 


38-94 


\ 


298-7 


6866-1 


47-68 


* 


266-2 


564 


11 


89-07 


I 


294-5 


6902-9 


47 93 


85 


267-0 


567 


4-5 


89-40 


94 


295-3 


6939-7 


48-19 


i 


267-8 


5707-9 


89-68 


i 


296-0 


6976-7 


48-45 


i 


268-6 


5741-4 


39-87 


\ 


296-8 


7013-8 


48-70 


f 


269-3 


5775-0 


40-10 


f 


297-6 


7050-9 


48-96 


86 


270-1 


5808-8 


40-83 


95 


298-4 ! 


7088-2 


49-22 


i 


270-9 


5842-6 


40-57 


i 


299-2 


7125-5 


49-48 


i 


271-7 


5876-5 


40-80 


\ 


800-0 


7163-0 


49-64 


i 


272-5 


5910-5 


4104 


\ 


800-8 


7200-5 

1 


50 00 

i 
1 



166 



ME>fSURATION OF ClRCLES. 




















Diam. 


Circum. 


Area in 


Area in 


! 

Diam. 


Circum. 


Area in 


Area in 


iu. 
96 


inches. 


square in. 


square ieet. 


121 


inche.s. 


square in. 


square feet. 


301-5 


7238-2 


50-26 


380-1 


11499-0 


79-85 


i 


302-3 i 


72759 


50-52 


122 


383-2 


11689-9 


81-18 


i 


303-1 


7313-8 


50-78 


1-23 


386-4 


11882-3 


82-51 


i 


303-9 


7351-7 


51-05 


1 1-24 


389-5 


12076-3 


83-86 










\ 125 


392-7 


1-2271-8 


85-22 


97 


304-7 


7389-8 


51-35 










I 


305-5 
306-3 
307 


74-27-9 
7466-2 
7504-5 


51-57 
51-84 
52-11 


126 
127 
128 


395-8 
398-9 
402 1 


12469-0 
12667-7 
12867-9 


86-59 
87-97 
89 36 


98 


307-8 


7542-9 


52-38 


1-29 
; 130 


405-2 

408-4 


13069 8 
13273-2 


90-76 
92-17 


i 


308-6 


7581-5 


52-65 










i 


309-4 


7620-1 


52-91 










f 


310-2 


7658-8 


53-18 


Il31 


411-5 


13478-2 


92-59 










132 


414-6 


13684-8 


95 03 


99 


311 


7697-7 


53-45 


133 


417-8 


13892-9 


96-47 


i 


311-8 


7736-6 


53-72 


134 


420-9 


14102-6 


97 93 


i 


312-5 


7775-6 


53-99 


135 


424-1 


14313-9 


99-40 


f 


313-3 


7814-7 


54-26 










100 


3141 


7854-0 


54-54 


136 


427-2 


14526-7 


100-88 


101 
102 
103 
104 


317-3 

320-4 
323-5 
326-7 


8011-7 
8091-2 
8332-3 
8494*9 


55-63 
5674 
57-86 
58-99 


' 137 

I 138 
1 139 
1 140 


430-3 
433-5 
436-6 

439-S 


14741-1 
14957-1 
15174-7 

15393-8 


102-36 
103-87 
105-37 
106-90 


105 


329-8 


8659-0 


60-13 










I 
1 










141 


442-9 


15614-5 


108-43 


106 


333-0 


8824-7 


61-28 


1142 


446-1 


15836-8 


109-97 


107 


336-1 


8992-0 


6244 


143 


449-2 


16060-6 


111-53 


108 


339-2 


9160 9 


63-61 


; 144 


452-3 


16286-0 


11309 


109 


342-4 


9381-1 


64-80 


i 145 


455-5 


16513 


114-67 


110 

! 


345-5 


9503-3 


65-99 


1 










lit- 

112 
1 113 
' 114 

115 


3487 
351-8 
3550 
358-1 
361-2 


9676-9 

9852-0 

10028-7 

10207-0 

10386-9 


67-20 
68-41 
69-64 

70-88 
72-13 


il46 

|147 

|148 

149 

|150 


458-6 
461-8 
464-9 
468-0 
471-2 


16741-5 
16971-7 
17-203-4 
17436-6 
17671-5 


116 
117 
119 
121 
122 


-26 

•86 
•46 
-08 
•71 


116 


364-4 


10568-3 


73-39 


j 151 


474-3 


17907-9 


124-36 


117 


367-5 


10751-3 


74-66 


! 152 


477-5 


18145-9 


126-01 


118 


370-7 


10935-9 


75-94 


i 153 


480-6 


18385-4 


127-67 


119 


373-8 


11122-0 


77-23 


I 154 


483-8 


18626-5 


129-35 , 


120 


376-6 


11309-7 


78 54 


I 155 


486-9 


18869-2 


131 


-03 \ 



Circumferences and Areas of Circles. 



167 



TABLE 
Of the Circumferences and Areas of Circles^ from 1 io hO feety 
advancing hy art inch. 



Diam, 


Circumference 




Diatn. 


Circumference 




ft. & in. 


in feet and in. 


Area in feet. 


ft. & in. 


in feet and in. 


Area in feet. 


1/^- 


8 


If 


•7854 


8 


13 4i 


14-1862 


1 


3 


4i 


•9217 


4 


13 7i 


147479 


2 


8 


8 


1-0690 


5 


13 10^ 


15-3206 


3 


3 


11 


1-2271 


6 


14 If 


16-9043 


4 


4 


2i 


1-3962 


7 


14 4f 


16-4986 


5 


4 


H 


1-5761 


8 


14 7i 


17-1041 


6 


4 


H 


1-7671 


9 


14 11 


17-7205 


1 


4 


lif 


1 9689 


10 


15 2i 


18-3476 


8 


6 


2f 


2-1816 


11 


15 5i 


18-9858 


9 


6 


n 


2-4052 


5 A 


15 8-i- 


19-6350 


10 


5 


9 


2-6398 


1 


15 llf 


20-2947 


11 


6 


2i 


28852 


2 


16 2f 


20-9656 


2A 


6 


3| 


3-1416 


3 


16 5f 


21-6475 


1 


6 


6i 


3-4087 


4 


IQ 9 


22-3400 


2 


6 


n 


3-6869 


5 


17 01 


230437 


3 


7 


Of 


3-9760 


6 


17 3J 


23-7583 


4 


7 


8i 


4-2760 


7 


17 ^ 


24-4835 


5 


7 


7 


4-5869 


8 


17 9f 


25-2199 


6 


7 


lOJ 


4-9087 


9 


18 Of 


25-9672 


7 


8 


If 


5-2413 


10 


18 3i 


26-7251 


8 


8 


4i 


5-5850 


11 


18 7i 


27-4943 


9 


8 


Vf 


5-9395 


6/^. 


18 10| 


28-2744 


10 


8 


I Of 


6 3049 


1 


19 li 


29-0649 


11 


9 


ll 


6-6813 


2 


19 4f 


29-8668 


3 A 


9 


5 


7-0686 


3 


19 7i 


30-6796 


1 


9 


8i 


7-4666 


4 


19 lOf 


31-5029 


2 


9 


llf 


7-8757 


5 


20 n 


32-S&76 


3 


10 


2i 


8 2957 


6 


20 H 


83-1831 


4 


10 


5f 


8-7265 


7 


20 81 


340891 


6 


10 


8f 


9-1683 


8 


20 Hi 


34-9065 


6 


10 


n| 


96211 


9 


21 2| 


35-7847 


7 


11 


8 


10-0846 


10 


21 6i 


36-6735 


8 


11 


6i 


10-5591 


11 


21 8f 


37-5736 


9 


11 


9S 


11 0446 


nft. 


21 ll| 


38-4 846 


10 


12 


5i 


11-5409 


1 


22 3 


394060 


11 


12 


3f 


12-0481 


2 


22 61 


40-8888 


4A 


12 


6f 


12-5664 


3 


22 9i 


41-2825 


1 


12 


91 


13-0952 


4 , 


23 Of 


42-2867 


2 


13 


1 


13-6353 


5 1 
1 


23 21 


43-2022 



J 68 


CiRCUMFERENCKS AND ArEAS 


OF Circles 


. 


Diam. 


Circumrerence 




Diam. 


Circumference 




ft. & in. 


in leet and in. 


Area in feet. 


ft. & in. 


in feet and in. 


Area in feet. 


\ 6 


23 6f 


44-1787 


3 


45 4i 


99-4021 


1 7 


23 11 


451656 


4 


35 7i 


100-8797 


8 


24 H 


46-1638 


5 


35 lOf 


102-3689 


9 


24 4i 


47-1730 


6 


36 li 


1U3-8691 


10 


24 1i 


481926 


7 


36 4i 


105-8794 


11 


21 lOf 


49-2286 


8 


36 7f 


106-9013 


Sft 


25 1^ 


50-2656 


9 


36 lOJ 


108-4342 


1 


25 4| 


51-3178 


10 


37 2f 


109-9772 


2 


25 1i 


52-3816 


11 


37 5i 


111-5319 


3 


25 11 


53-4562 


12 ft. 


37 8f 


113-0976 


4 


25 2i 


54-5412 


1 


37 IH 


114-6732 


5 


26 5i 


55-6377 


2 


38 2f 


116-2607 


6 


26 8f 


56-7451 


3 


38 5f 


117-85.90 


.1 


26 Hi 


57-8628 


4 


38 8^ 


119-4674 


8 


27 2f 


58 9920 


5 


39 


121-0876 


9 


27 5| 


60-1321 


6 


39 3i 


1227187 


10 


27 9 


61-2826 


7 


39 6f 


124-3598 


11 


28 Oi 


62-4445 


8 


39 n 


126 0127 


9/^. 


28 3i 


63-6174 


9 


40 Of. 


127-6765 


1 


28 6f 


64-8006 


10 


40 3f 


129-3504 


2 


28 9^ 


65-9951 


11 


40 6i 


131^0360 


3 


29 Of 


67-2007 


13/if. 


40 10 


132-7326 


4 


29 3| 


68-4166 


1 


41 li 


134-4391 


5 


29 7 


69-6440 


2 


41 4f 


136-1574 


6 


29 lOi 


70-8823 


3 


41 7i 


137-8867 


7 


30 li 


72-1809 


4 


41 lOf 


139-6260 


8 


30 4f 


73-3910 


5 


42 If 


141-3771 


i ^ 


30 7i 


74-6620 


6 


42 4f 


143-1391 


\ 10 


30 ni- 


75-9433 


7 


42 8 


144-9111 


1 11 


si If 


77-2362 


8 


42 Hi 


146-6949 


; lo.A 


31 5 


78-5400 


9 


43 2i 


148-4896 


1 


31 8i 


79-8540 


10 


43 5i- 


150-2943 


2 


31 111 


81-1795 


11 


43 8f 


152-1109 


3 


32 2f 


82-5160 


14/^. 


43 llf 


153-9384 


1 4 


82 5i 


83-8627 


1 


44 21- 


155-7758 


5 


32 8| 


85-2211 


2 


44 


6 


157-6250 


6 


32 llf 


86-5903 


3 


44 


H 


159-4852 


1 7 


33 2| 


87-9697 


4 


45 


Oi 


161-3553 


1 8 


33 6i 


89-3608 


5 


45 


H 


163-2373 


! 9 


33 9i 


90-7627 


6 


45 


6f 


165-1303 


1 10 


34 Of 


92-1749 


7 


45 


9f 


167-0331 


j 11 


34 SI 


93 5986 


8 


46 


Oi 


168-9479 


1 11 i^. 


34 61- 


95-0334 


9 


46 


4 


170-8735 


1 • 1 


34 9f 


96-4783 


10 


46 


7t 


172-8091 


t 2 

i 
1 


35 Oi 


97-9347 


11 


46 1 


Lli 


174-7565 



Circumferences and Areas of Circles. 



169 





Dium. 


Circumterenre 




Diam. 


Circumference 




ft. A; in. 


in feet and in. 


Area in feet. 


ft. & in. 


in feet and in. 


Area in feet. 


! 

Uft 


47 U 


176-7150 


9 


58 lOI 


2*76-1171 


1 


47 4f 


178-6832 


10 


59 2 


278-5761 


2 


47 7f 


180-6634 


11 


59 5i 


281-0472 


3 


47 lOj 


182-6545 


19/f. 


59 8i- 


283-5294 


4 


48 2^ 


184 6555 


1 


59 Hi 


286-0210 


5 


48 qI 


186-6684 


2 


60 2i 


288-5249 


6 


^^8 8i 


188-6923 


3 


60 6|- 


291-0397 


7 


48 Hi 


190-7260 


4 


60 8f 


293-5641 


8 


49 21- 


192-7716 


5 


60 111 


296-1107 


9 


49 5f 


194-8282 


6 


61 3i 


298-6483 


10 


49 81 


196-8946 


7 


61 6i 


301-2054 


11 


50 


198-9730 


8 


61 9i 


303-'7747 


nft. 


50 ^ 


201-0624 


9 


61 Oi 


306-3550 


* 1 


50 6i 


203-1615 


10 


62 3f 


308-9448 


2 


50 9f 


205-2726 


11 


62 6f 


311-5469 


3 


51 Oi 


207-3946 


20//:. 


62 91 


314-1600 


4 


51 3f 


209-5264 


1 


63 H 


316-7824 


6 


51 6i 


211-6703 


2 


63 4i 


319-4173 


6 


51 10 


213-8251 


3 


63 7t 


322-0630 


n 


52 li 


215-9896 


4 


63 Hi 


324-7182 


8 


52 4J 


218-1662 


5 


61 If 


327-3858 


9 


52 7f 


220-3537 


6 


64 4f 


330-0643 


10 


52 10^ 


222-5510 


7 


64 7f 


332-7522 


11 


53 If 


224-7603 


8 


64 11 


335-4525 


\1 ft. 


53 4i 


226-9806 


9 


65 2i 


338-1637 


1 


53 8 


229-2105 


10 


65 5| 


340-8844 


2 


53 Hi 


231-4525 


11 


65 Si 


343-6174 


3 


54 2i 


233-7055 


21 //. 


65 llf 


346-3614 


4 


54 5| 


235-9682 


1 


66 2f 


349-1147 


5 


54 %\ 


238-2430 


2 


66 5| 


351-8804 


6 


54 IH 


240-5287 


3 


66 9 


354-6571 


7 


55 2J 


2428241 


4 


66 OJ 


357-4432 


8 


55 6 


245-1316 


5 


67 3t 


360-2417 


9 


55 94 


247-4500 


6 


67 6i 


363-0511 


10 


56 Oi 


249-7781 


7 


67 H 


365-8698 


11 


56 3i 


252-1184 


8 


68 Of 


368-7011 


18//. 


56 6i 


254-4696 


9 


68 3J 


371-5432 


1 


56 9|- 


256-8303 


10 


68 7 


374-3947 


2 


57 Oi 


259-2033 


11 


68 lOi 


377-2587 


3 


57 4 


261-5872 


22 ft. 


69 If 


380-1336 


4 


57 71 


263-9807 


' 1 


69 4j 


383-0177 


5 


57 lOi 


266-3864 


2 


69 7f 


385-9144 


6 


58 If 


268-8031 


3 


69 lOf 


888-8220 


7 


58 4i 


271-2298 


4 


70 IJ 


391 7389 


8 


58 7f 


273 6678 


5 


70 5 


394-6683 



15 



no 



Circumferences and Areas of Circles. 



i 


i Diana. 


Circumference 






Diam. 


Circumference 




j ft.&in. 


in feet 


dnd in. 


Area 


m feet. 


ft. & in. 


m feet and in. 


Area m feet. 


6 


10 8i 


397-6087 


3 


82 6i 


541-1896 


7 


10 Hi 


; 400-5583 


4 


82 8| 


644-6299 


8 


^71 2i 


i 403-5204 


5 


82 llj 


648 0830 


9 


VI H 


i 406-4935 


6 


83 3 


651 -5471 


10 


11 8f 


409-4759. 


7 


83 61 


555 0201 


11 


71 llj 


412 


-4707 


8 


83 9i 


658 5059 


23 ft 


72 3 


415 


-4766 


9 


84 Of 


562-0027 


1 


72 6i 


418 


•4915 


10 


84 3i 


665-5084 


2 


72 9f 


421 


-5192 


11 


84 6f 


669-0270 


3 


73 Oi 


424 


•5577 


21 ft. 


84 9| 


672-5566 


4 


73 3f 


427 


-6055 


1 


85 1 


576-0949 


6 


73 6f 


430 


•6658 


2 


85 4i 


679-6463 


6 


73 9| 


433 


•7371 


3 


85 8i 


683-2085 


1 


74 1 


436 


•8175 


4 


85 11| 


686'7796 


8 


74 4i 


439 


•9106 


5 


86 li 


690'3637 


9 


74 n 


443 


-0146 


6 


86 4i 


593-9687 


10 


74 m 


446 


-1278 


7 


86 7| 


697 5625 


11 


75 


H 


449 


-2536 ! 


8 


86 11 


601 1793 


24.ft 


75 


4f 


452 


3904 


9 


87 2i 


604 8070 


1 


75 


H 


4r>5 


-5362 


10 


87 5J 


608-4436 


2 


75 1 


1 


458 


•6948 


11 


87 8f 


6120931 1 


3 


76 


2i 


461 


•8642 


28 ft. 


87 in 


615 7536 


4 


76 


H 


465 


0428 


1 


88 2f 


6194228 


5 


76 


H 


468 


2341 


2 


88 5f 


623-1050 


6 


76 1 


H 


471 


•4363 


3 


88 9 


626^7982 


7 


77 


2f 


474 


6476 \ 


4 


89 


0^ 


630-5002 


8 


77 


5i 


477 


8716 1 


5 


89 


Si 


634^2152 


9 


77 


9 


481 


1065 ! 


6 


89 6f 


637-9411 


10 


78 


Oi 


484 


3506 


7 


89 U 


641 •6758 


11 


78 


H 


487 


6078 i 


8 


90 Of. 


645^4235 


26ft. 


78 


H 


490 


8750 1 


9 


90 


3f 


6491821 


1 


78 


n 


494 


1516 : 


10 


90 


H 


652-9496 


o 


79 


01 


497 


4411 ; 


11 


90 ] 


^H 


656^7300 


3 


79 


n 


500 


7415 


29 ft 


91 


H 


660-5214 


4 


79 


7i 


504 


0510 


1 


91 


4f 


664 3214 


5 


79 Hi 


607-3732 ■ 


2 


91 


n 


668-1346 


6 


80 li 


510-7063 ! 


3 


91 ] 


[Of 


671-9587 


7 


80 4t 


514 


0484 ! 


4 


92 


If 


675-7915 


8 


80 7f 


617 


4034 1 


5 


.92 


4| 


679^6375 


9 


80 lOf 


620 


7692 


6 


92 


H 


683-4943 


10 


81 n 


524 


1441 


7 


92 ] 


n 


687-3598 


11 


81 5 


527 


5318 


8 


93 2| 


691-2385 


26//. 


81 


8i 


530 


•9304 


9 


93 5i 


695^1280 


1 


81 Hi 


534 


3379 


10 


93 81- 


699-0'263 


2 


82 2f 


537 


•7583 


11 


93 h| 


702^9377 



Circumferences and Areas of Circles. 



171 





Diam. 


Circumference 




Diiim. 


Circumference 




ft.&in. 


in feet and in. 


Area in feet. 


ft. <fc in. 


in feel and in. 


Area in feet. 


30/^. 


94 2| 


706-8600 


9 


106 Oi 


894-6196 


1 


94 6 


710-7909 


10 


106 3f 


8y9 0413 


2 


91 Si 


714-7350 


11 


106 6t 


903-4763 


3 


95 9f 


718-6900 


Uft 


106 9f 


907-9224 


4 


95 3| 


722-6537 


1 


107 0| 


912-3767 


5 


95 6f 


726-6305 


2 


107 4 


916-8445 


6 


95 9| 


730-6183 


3 


lu7 7i 


921-3232 


7 


96 Oi 


7346147 


4 


107 lOi 


925-8103 


8 


96 4 


738-6242 


5 


108 ]f 


930-3108 


9 


96 7i 


742-6417 


i 6 


108 4|- 


934-8223 


10 


96 10§ 


746-6738 


1 '^ 


108 7f 


939-3421 


11 


97 H 


7507161 


i 8 


108 10| 


943-8753 


31/^ 


97 4f 


751-7691 


i 9 


109 2 - 


948-4195 


1 


97 7f 


758-8311 


10 


109 51 


952-9720 


2 


97 10| 


7629062 


11 


109 8i 


957-5380 


3 


98 2 


766-9921 


85//. 


109 llf 


962-1150 


4 


98 5^ 


771-0866 


1 


110 2| 


966 70O1 





98 8f 


775-1914 


2 


110 5f 


9712989 


6 


98 IH 


779-3131 


3 


110 84 


975-9085 


7 


99 2f 


783-1403 


4 


111 


980-5264 


8 


99 5f 


787-5808 


5 


111 31 


985-1579 


9 


99 8J 


791-7322 


6 


111 6J 


989-8003 


10 


100 


795-8.)22 


7 


111 9f 


994-4509 


11 


100 3^ 


800-0654 


8 


112 Oi 


999-1151 


32//. 


100 6f 


804-2196 


9 


112 3f 


1003-7902 


1 


100 91- 


808-4422 


10 


112 6i 


1008-4736 


2 


101 0| 


812-6481 


11 


112 10 


1013 1705 


3 


101 3| 


816-8650 


36// 


113 ]| 


1017-8784 


4 


101 6J 


821-0904 


1 


113 4i 


1022-5944 


5 


101 10 


825-3291 


2 


113 7| 


1027-3-240 


6 


102 li 


829-5787 


3 


113 lOf 


1032-0646 


7 


102 4f 


833-8368 


4 


114 If 


1036-8134 


8 


102 7i 


838-1082 


5 


114 4| 


1041-5758 


9 


102 101 


8123905 


6 


114 8 


1046-3491 


10 


10:^ If 


846-6813 


7 


114 111 


1051 1306 


11 


103 4J 


850-9855 


8 


115 2i 


1055-9257 


33//. 


103 8 


855-3006 


9 


115 5| 


1060-7317 '• 


1 


103 11| 


859-6240 


10 


115 9i 


1065-5459 1 


2 


104 2i 


863-9609 


11 


115 llf 


1070-3738 ■. 


3 


104 5S 


868-3087 


37 //. 


116 2i 


1075-2126 


4 


104 8f 


872-6649 


1 


116 6 


1080-0594 ' 


6 


104 llf 


877-0346 


2 


116 91 


1084-9201 


6 


105 2| 


881-4151 


8 


117 Oi 


1089-7915 


7 


105 6 


885-8040 


4 


1 17 3^ 


1094-6711 


8 


105 9| 


890-2064 


5 


117 &i 


1099-5644 



172 



Circumferences and Areas ok Circles. 















Diam. 


Circumference 




Diam. 


1 
Circumference 




ft «fein. 


in leet and in. 


Area in feet. 


fi. & in. 


in feet and in. 


Area in feet. 


6 


117 9f 


1104-4687 


3 


129 7 


1336-4071 


1 


118 Of 


1109-3810 


4 


129 lOi 


1341-8101 


8 


118 4 


1114-3071 


5 


130 If 


1347-2271 


9 


118 71 


1119-2440 


6 


130 4i 


! 1352-6551 


10 


118 lOi 


1124 1891 


7 


130 71- 


1358-0908 


11 


119 If 


1129-1478 


8 


130 lOf 


1363-5406 


38 /t 


119 4i 


1134-1176 


1 9 


131 n 


1369-0012 


1 


119 7i 


1139-0953 


1 10 


131 5 


1374-4697 


2 


119 lOf 


1144 0868 


1 11 


131 8i 


1379-9521 


3 


120 2 


1149-0892 


1 42 ft. 


131 llf 


1385 4456 


4 


120 5 


1154-0997 


1 


132 2i- 


1390-2467 


5 


120 8f 


1159-1239 


i 2 


132 5i 


1 1396-4619 


6 


120 llf 


1164-1591 


i 3 


132 8f 


1 1401-9880 


7 


121 U 


1169-2023 


4 


132 llf 


1 1407-5219 


8 


121 of 


1174-2592 


5 


133 3 


i 1413-0698 


9 


121 8f 


1179-3271 


6 


133 6i 


1418-6287 


10 


121 U-l 


1184-4030 


7 


133 9? 


1424-1952 


11 


122 31 


1189-4927 


8 


134 Oi 


1429-7759 


39 /if. 


122 6i 


1194-5934 


9 


134 8f 


1435 3675 


1 


122 9i 


1199-7195 


: 10 


134 6f 


1440-9668 


2 


123 Oi 


1204 8244 


! 11 


134 9f 


1446-5802 


3 


123 3f 


1209-9577 


43 ft 


135 1 


1452-2046 


4 


123 6f 


1215-0990 


1 


135 4f 


1457-8365 


5 


123 9i 


1220-2542 


2 


135 7i 


1463-4827 


6 


124 11 


1225-4203 


3 


135 10| 


1469-1397 


7 


124 4i 


1230-5943 


4 


136 If 


14748044 


8 


124 7| 


1235-7822 


5 


136 4f 


1480-4833 


9 


124 lOi 


1240-9810 


6 


136 7f 


1486 1731 


10 


125 H 


1246-1878 


7 


136 11 


1491-8705 


11 


125 4f 


1251-4084 


i 8 


137 2f 


1497-5821 


40/^. 


125 7i 


1256-6400 


9 


137 5^ 


1503-3046 


I 


125 11 


1261-8794 


10 


137 8f 


1509-0348 


2 


126 2i 


1267-1327 


11 


137 llf 


1514-7791 


3 


126 of 


1272 3970 


44/^. 


138 2f 


1520-5344 


4 


126 8i- 


1277-6692 


1 


138 ^ 


1526-2971 


5 


126 llf 


1282-9553 


2 


138 9 


15320742 


6 


127 2f 


1288-2523 


3 


139 Of 


1537-8622 


7 


127 5| 


1293-5572 


4 


139 3i 


1543-6578 


8 


127 9 


1298 8760 


5 


139 6f 


1549-4776 


9 


128 Oi i 


1304-2057 


6 


139 9f 


1555-2883 


10 


128 3f i 


1309-5433 


7 


140 Of 


1561-1165 


11 


128 6i 


1314-8949 


8 


140 3f 


15669591 


4.1 ft. 


128 9f 


1320-2574 


9 


140 7f 


1572-8125 


1 


129 Of 


1325-6276 


10 


140 lOf 


1578-6735 


2 


129 3f 


1331-0119 


11 


141 li 


1584-5488 



Circumferences axd Areas of Circles. 



IIS 



Diam. 
ft. & in. 



Uft. 
1 
2 
3 
4 
5 
6 
1 
8 
9 
10 
11 
46/f. 
1 
2 
3 
4 
5 
6 
1 
8 
9 
10 
11 
41ft. 
1 
2 
8 
4 
5 
6 



Circumference 
in feet and in. 



141 4f 

141 7i 

141 lOf 

142 11 
142 5 
142 8i 

142 Hi 

143 2f 
143 5i 
143 8f 

143 11| 

144 3 
144 6i 

144 H 

145 Of 
145 3| 
145 6f 

145 9i 

146 li 
146 4i 
146 Vi 

146 lOf 

147 H 
147 4f 
147 7f 

147 11 

148 2i 
148 5i 
148 8f 

148 Hi 

149 21- 



Area in feet. 



1590-4350 
1596-3286 
1602 2366 
1608-1555 
16140819 
1620-0226 
1625-9743 
lH3 1-9834 
1637-9068 
1643-8912 
1649-8831 
1655-8892 
1661-9061 
1667-9308 
1673-9698 
1680-0196 
1686-0769 
1692-1485 
1698 2311 
1704-3210 
17104254 
17165407 
1722-6634 
1728-8005 
1734-9486 
1741-1039 
1747-2738 
1753-4545 
1759-6426 
1765-8452 
1772-0587 



Diam. 
ft. & in. 



10 
11 

48/^. 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 

49/^. 
1 
2 
3 
4 
5 
6 
7 



10 

11 

50/^. 



Circumference | 


in leet 


and in. 


149 


5| 


149 


8* 


l/.O 


Oi 


150 


H 


150 


H 


150 


H 


151 


0* 


151 


3f 


151 


H 


151 


m 


152 


n 


152 


4f 


152 


7^' 


152 


10* 


153 


If 


153 


4* 


153 


H 


153 


Hi 


154 


2| 


154 


H 


154 


H 


154 


m 


155 


n 


155 


6 


155 


H 


156 


Oi 


156 


H 


156 


6f 


156 


9f 


157 


Oi 



Area in feet. 

17782795 

1784-5148 

1790-7610 

1797-0145 

1803-2826 

1&09-5616 

1815-8477 

1822 1485 

18284602 

1834 7791 

1841-1127 

1847-4571 

1853-8087 

1860-1750 

1866 5521 

18729365 

1879-3355 

1885-7454 

1892-1724 

1898-5041 , 

1905 0367 

1911-4965 

1917 9609 

1924-4263 

1930-9188 

1937-3159 

1943-9140 

1950-4392 

1956-9691 

1963-5000 



To Preserve Steel Goods. — Caoutchouc 1 part; turpentine 16 
parts. Dissolve with a gentle heat, then add boiled oil 8 parts. 
Mix by bringing them to the heat of boiling water ; apply it to the 
steel with a brush, in the way of varnish. It may be removed with 
turpentine. The oil may be wholly omitted. 

Size. — Oil size is made by grinding yellow ochre or burnt red 
ochre with boiled linseed oil, and thinning it with oil of turpentine. 
Water size (for burnished gilding) is parchment size ground with 
yellow ochre. 

Silica and Carbon. — Silica is the base of the mineral world. 
Carbon the base of the organized. 



15* 



174 Ivory. 



IVOEY. 

How to Soften it. — Take 3 oz. spirits of nitre, and 15 of spring 
water; mix together; drop in the Ivor^^ and let it soak. In three 
or four days it will be so soft as to obe}' your fingers. 

How to Dye Ivory vihen Softened. — If you desire to dye Ivory 
when thus softened, dissolve, in spirits of wine, such colors as you 
wish to use. When the spirit of wine is sufficiently tinged with 
the color yoa have put in plunge in your Ivory, and leave it there 
till it is dyed to suit you. Then take out the Ivory and give it 
what form you please. 

How to Harden Ivory. — To harden the Ivory afterwards, wrap it 
up in a sheet of white paper, cover it with dry, decrepitated salt, 
and lay it by for twenty-four hours, Avhen it will be restored to its 
original hardness. 

To re- Wliiten loory lohich has Turned a Broicn Yellow. — ^There are 
two ways of doing this, namely : 1. Slack some lime in water, into 
which drop the ivory ; decant it gently, and boil till it looks quite 
white. 2. To polish it afterwards, set it in the turner's wheel, and 
after having worked it, take some rushes and pumice stone, mix a 
subtile powder with water, and rub till it becomes perfectly smooth : 
then heat it b}^ turning it over a piece of linen or sheepskin, and 
when hot rub it with a little whitening diluted with olive oil ; then 
rub it with a little dry whitening alone, and finally with a piece 
of soft Avhite rag, and the Ivory will look remarkably white. 

How to Dye Ivory Black. — Immerse the Ivory in a boiling solution 
of logwood, then take it out, and wash it in a solution of copperas. 

Bine. — There are two ways of reaching this color. The first is 
to soak the Ivory in a solution of verdigris in nitric acid, which 
will make it green; then dip it into a solution of boiling hot 
pearlash, and it will turn blue. The second way is as follows: 
Immerse the Ivor}^ in a solution of sulphate of indigo and water, 
partly neutralized with potash. 

Green. — Steep blued Ivory in a solution of nitro-muriate of tin, 
and then in a decoction of fustic. Another and a more instanta- 
neous plan is to immerse it in a solution of acetate of copper. 

Yellow. — Steep the Ivory in a bath of neutral chromate of potash, 
and afterwards in a boiling solution of acetate of lead. 

Red. — Soak the Ivory for a short time in a solution of tin, and 
then in a decoction of cochineal. 

Violet. — Moisten the Ivory with a solution of tin, as before ; then 
immerse it in a decoction of logwood. 

Purple. — Soak the Ivory in a solution of sal ammoniac into four 
times its weight of nitrous acid. 

Fluid for MarJcivg Ivory. — Take nitrate of silver, 2 parts ; nitric 
acid, 1 part; water, 7 parts. Mix. 

Etching Fluid for Ivory. — Take of diluted sulphuric acid and 
diluted muriatic acid, equal parts. Mix. 



Centre. l*?© 

Etching Varnish for Ivory. — White wax, 2 parts , tears of mastic, 
2 parts. Mix. 

To Gild Ivory. — Immerse it in a solution of nitro-muriate of gold, 
and then, while yet damp, expose it to hydrogen gas. Wash it 
afterwards in clean water. Another plan of gilding Ivory is by 
immersing it in a fresh solution of proto-sulphate of iron, and 
afterwards in a solution of chloride of gold. 

To Polish Ivory. — Use a rubber and putty and water. 

The hardest, toughest, whitest, and most translucent ivory has 
the preference in the market; and the tusks of the sea-horse are 
considered to afford the best. Ivory has the same constituents as 
the teeth of animals : three-fourths being phosphate, with a little 
carbonate of lime ; one-fourth cartilage. With regard to dyeing 
Ivory, it may in general be observed, that the colors penetrate 
better before the surface is polished than afterwards. Should any 
dark spots appear, they may be cleared up by rubbing them with 
chalk; after which the Ivory should be dyed once^iore to produce 
a perfect uniformity of shade. On taking it out of the boiling hot 
dye bath, it should be plunged immediately into cold water, to 
prevent the chance of fissures being caused by the heat. 



CENTEE, 



In a general sense, denotes a point equally remote from the 
extremes of a line, surface, or solid. 

Centre of Attraction 

Of a body, is that point into which if all its matter is collected, 
its action upon any remote particle would still be the same. 
Centre of Equilihriuin 

Is the same, in respect to bodies immersed in a fluid, as the cen- 
tre of gravity is to bodies in free space. 

Centre of Friction 

Is that point in the base of a body on which it revolves, into 
which if the whole surface of the base and the mass of the body 
were collected, and made to revolve about the centre of the base 
of the given body, the angular velocity destroyed by its friction 
would be equal to the angular velocity destroyed in the given body 
by its friction in the same time. 

Centre of Gravity 

Of any body, or s^^stem of bodies, is that point upon which the 
body, or system of bodies, acted upon only by the force of gravity, 
will balance itself in all positions ; hence it follows, that, if a line 
or plane, passing through the centre of gravity, be supported, the 
body or system will be also supported. 



Pfi Cohesion". 



Centre of Gyration 

Is that point into which, if the whole mass were collected, a given 
force, applied at a given distance, would produce the same angular 
velocity in the same time as if the bodies were disposed at their 
respective distances. 

This point differs from the Centre of Oscillation only in this, 
that, in the latter case, the motion is produced by the gravity of 
the body; but, in the former, the body is put in motion by some 
other force, acting at one place only. 



COHESION 



Is that species of attraction which, uniting particle to particle, 
retains together the component parts of the same mass ; being thus; 
distinguished from Adhesion, or that species of attraction which- 
takes place between the surfaces of similar or dissimilar bodies, j 
The absolute cohesion of solids is measured by the force necessary i 
to pull them asunder. Thus, if a rod of iron be suspended in a| 
vertical position, having weight attached to its lower extremity! 
till the rod breaks, the whole weight attached to the rod, at thei 
time of fracture, will be the measure of its cohesive force, or abso-! 
lute cohesion. 1 

The particles of solid bodies, in their natural state, are arranged j 
in such a manner, that they are in equilibrium in respect to the! 
forces which operate on them; therefore, when any new force is ;^ 
applied, it is evident that the equilibrium will be destroyed, and; 
that the particles will move among then]selves till it be restored.! 
When the new force is applied to pull the body asunder, the body; 
becomes longer in the direction of the foice, which is called thei 
extetision ; and its area, at right angles to the direction of the force,! 
contracts. When the force is npplied to compress the body, itj 
becomes shorter in the direction of the force, wliich is called the; 
compression; and the area of its section, at right angles to the! 
force, expands. In either case, a part of the heat, or any fluid that' 
occupies the pores or interstices of the body, belore the new force i 
was made to act upon it, will be expelled. I 

Platin*a-Mohr. — Zinc two parts : platinum one part. Melt:Ji<l. 
i reduce the alloy to powder, which must be treated with dilute 
I sulphuric acid until all the zinc is washed out ; then wash it witli 
I water, digest it in a ley of potash, and again wash it with water, i 
I This powder possesses the property of converting alcoiiol into: 
, vinegar. : 

' The Velocity of SorxD. — It has been ascertained, by careful; 
I investigation, that sound passes in water at a speed of 4,708 feet j 
per second. 



Laws of Elastic Fluids. 1'?7 



MECHANICAL LAWS OF ELASTIC FLUIDS. 

Boyle s or Mariotte's Law. * 

The elastic force of a gas or air at a given temperature is 
inversely proportional to the space which it occupies. 

Let p =r elastic force of a gas when it occupies the space s. 
P= do. do. S. 

The elastic force of any gas at a given temperature is propor- 
tional to its density. 

The density of any body is the weight of a cubic unit of it, 
usually one cubic foot. 

Let p — the elastic force when the density is d. 

And A; — do. do. unit. 

Dalton^a and Gay-Lussaca Law. 
All gases, under the same pressure, undergo equal expansions 
foi equal increments of temperature. 

It was ascertained by these eminent philosophers, that 100 
measures of air expand to 137-5 measures on being heated from 
32° to 212° of Fahrenheit's thermometer, hence 

37-5 =r increments of 100 measures for 180 degrees of heat 

37 '5 

35-- do. 1 180 do. 

•375 
-f^^ do. 1 1 do. 



480 



l^et V — volume of any gas at the temperature t. 
V= da do. i\ 

Then, V = ^-1-^^-1=^) -F accurately. 
\ ^ a{t — 32) ^ 

= I 1 + a (^'— <) Iv very nearly. 

Amonton^s Lmio. 
This law is the relation between the elastic force, the density, 
and the temperature, of any gas. If, then, the volume of a gas be 
constant, its elastic force will increase ; and, if the elastic force is 
constant, its volume will increase for every increase of tempera- 
ture. It is important to connect these quantities by an equation. 



"1'78 Laws of Elastic Fluids. 



Put p rr= elastic force of a ^as at the temperature Q° and density c?.' 
Then, p=zkd(\ + ai)) 
where A: is a constant quantit.y dependinjcr on the nature oi the gas, 
and a = -^\-^. 

When a light and heavy gas are once mixed, they do not exhibit 
any tendency to separate ; in this respect they differ from mixed 
liquids. 

Dalton's Experiment. \ 

The vessel a contains a light gas, as hydrogen, aiid 

the vessel b contains a heavy gas, as carbonic acid; tlie 

two gases are allowed to communicate by a narrow 

, tube c, an interchange speedil}^ takes place of a part 

if^ of their contents, which their relative position might 

' • be supposed would prevent. Contrary to gravity, the 

heavy gas ascends, and the light one descends, till, in a 
few hours, the gases become perfectly mixed, and the 
proportion of the two gases is the same in both vessels. 



Gases diffuse into the atmosphere and into each other with 
different degrees of rapidity. The velocity with which air will 
rush into a vacuum is 1 348 feet per second. 

To determine the velocity with which the air of the atmosphere 
will rush into a space containing rarer air : 

Let V = velocity of air, of density {d), rushing into a void. 

V= velocity of air rushing into air of density D. 



— <-?) 



There will always be a current so long as (B) and {d) are 
unequal. 

Illuminating Gases. 

Pure hydrogen burns with too feeble a flame to be employed for 
the purpose of illumination. Carburetted hydrogen has the pro- 
perty of pi'ecipitating its carbon; in the act of burning, its solid 
particles become incandescent, and diffuse a strong light. Tlie more 
carbon the gas contains the more brightly does it burn. 

Two measures of h3'drogen gas, with one measure of the vapor 
of carbon, form the carburetted hydrogen found in coal mines, and 
is also evolved in ditches, from decomposins: vegetable matter. 
Another kind of carburetted hydrogen, called olefiant gas, is 
formed by two measures of hydrogen and two measures of gaseous 
carbon. This gas burns with a brighter flame than the common 
carburetted hydrogen. 

The best substances for furnishing a gas rich in luminiferous 
materials are pit coal, resin, oil, fats of all kinds, tar, wax, <fec. 

The volume of gas discharged from the end of a pipe is directly 
proportional to the square of its diameter, and inversely as the 
square root of its length. 



Laws of Elastic Fluids. 1*79 



Let n = number of cubic feet of gas discharged per hour through I 
a length of pipes / feet and diameter D. i 

vr 

This formula is applicable only when the gas is transmitted 
through the pipes, without being let off in its way by burners. If 
the main send off branches for burners, then, for "the same length, 
the difinieter may be reduced ; or, for a like diameter, the length 
may be increased. 



Stains, to Remove. — Stains of iodine are removed by rectified 
spirit. Ink stains by oxalic acid or superoxalate of potash. Iron 
moidih by the same ; but if obstinate, it has been recommended to 
moisten them with itik, then remove them in the usual way. 

Red syots on black cloth, from acids, are removed by spirits of 
hartshorn, or other solutions of ammonia. 

Stains of Marking Ink, or Nitrate of Silver, to remove. 1. Wet 
the stain with fresh solution of chloride of lime, and after ten or 
fifteen minutes, if the marks have become white, dip the part in 
solution of ammonia or of hyposuJphite of soda. In a few minutes 
wash with clean water. 

2. Stretch the stained linen over a basin of hot water, and wet 
the mark with tincture of iodine. 

Browning, or Bronzing Liquids, for Gun Barrels. — 1. Aqua- 
fortis ^ oz., sweet spirit of nitre ^ oz., spirit of wine 1 oz., sulphate 
of copper 2 oz., water 30 oz., tincture of muriate of iron 1 oz. Mix. 

2. Sulphate of copper 1 oz,, sweet spirit of nitre 1 oz., water 1 
pint. Mix. In a few days it will be fit for use. 

3. Sweet spirit of nitre 3 oz., gum benzoin l^ oz., tincture of! 
muriate of iron ^ oz., sulphate of copper 2 dr., spirit of wine ^ oz. j 
Mix, and add 2 lbs. of soft water. j 

4. Tincture of muriate of iron -J oz., spirit of nitric ether ^ oz., i 
sulphate of copper 2 scruples, rain water -^ pint. The above are j 
applied with a sponge, after cleaning the barrel with lime and ! 
water. When drj^, they are polished with a stiff brush, or iron j 
scratch brush. | 

Bronzing Liquids for Tin Castings. — Wash them over, after; 
t>eing well cleaned and wiped, with a solution of 1 part sulphate of; 
iron, and 1 of sulphate of copper, in 20 parts of water; afterwards | 
with a solution of 4 parts verdigris in 11 of distilled vinegar: 
leave for an hour to dry, and then polish with a soft brush ai ' 
colcothar. 

SoLVRNTs for Gutta Percha — Bcuzole readily dissolves it: 
do chloroform and bisulphuret of carbon. 



180 Squahes, Cubes, Squaiik Root>> and Cube Roots. 



TABLE 

Of Squares, Cubes, Square and Cube Roots of Kumbers. 



Number. 


Square. 


1 


1 


2 


4 


3 


9 


4 


16 


5 


25 


6 


36 


n 


49 


8 


64 


9 


81 


10 


100 


11 


121 


12 


144 


13 


169 


14 


196 


15 


225 


16 


256 


17 


289 


18 


324 


19 


361 


20 


400 


21 


441 


22 


484 


23 


529 


24 


576 


25 


625 


26 


676 


27 


729 


28 


784 


29 


841 


30 


900 


31 


961 


32 


1024 


33 


1089 


34 


1156 


35 


1225 


36 


3 296 


37 


1369 


38 


1444 


39 


1521 


40 


1600 


41 


1681 



1 

8 

27 

64 

125 

216 

343 

512 

729 

1000 

1331 

1728 

2197 

2744 

3375 

4096 

4913 

5832 

6859 

8000 

9261 

10648 

12167 

18824 

15625 

17576 

19683 

21952 

24389 

27000 

29791 

32768 

35937 

39304 

42875 

46656 

50653 

54872 

59319 

64000 

68921 



Square Root. Cube Root. Number. 



1-0 

1-414213 

1-732050 

2-0 

2-236068 

2-449489 

2-645751 

2-828427 

3-0 

3 162277 

3-316624 

3-464101 

3-605551 

3-741657 

3-872983 

4-0 

4-123105 

4-242640 

4-358898 

4-472136 

4-582575 

4-690415 

4-795831 

4-898979 

50 

5-099019 

5-196152 

5-291502 

5-385164 

5-477225 

5-567764 

5-656854 

5-744562 

5-830951 

5-916079 

6-0 * 

6082762 

6164414 

6-244998 

6-324555 

6-403124 



1-0 

1-25992 
1-44225 
1-58740 

1 70997 
1-81712 
1-91293 
2-0 

2-08008 
215443 
3-22398 
2-28942 

2 35133 
2-41014 
2-46621 
2-51984 
2-57128 
2-62074 
2-66840 
2-71441 
2-75892 
2-80203 
2-84386 
2-88449 
2-92401 
2-96249 
3-0 

8-03658 
8-07231 
3-10723 
3-14188 
3-17480 
3-20753 
3-23961 
3-27106 
3-30192 
3-33222 
3-36197 
3-89121 
3-41995 
3-44821 



1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 





Squares, Cubes, Square Roots, and Cube Roots 


181 


Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Number. 


42 


1764 


74988 


6-480740 


3-47602 


42 


43 


1849 


79507 


6-557438 


3-50339 


43 


44 


1936 


85184 


6-633249 


3-53034 


44 


45 


2025 


91125 


6-708203 


3-55689 


45 


46 


2116 


97336 


6-782330 


3-58304 


46 


47 


2209 


103823 


6-855654 


3-60882 


47 


48 


2304 


110592 


6-928203 


8-63424 


' 48 


49 


2401 


117649 


7-0 


3-65930 


49 


50 


2500 


125000 


7-071067 


3-68403 


50 


51 


2601 


132651 


7-141428 


3-70842 


51 


52 


2704 


140008 


7-211102 


3-73251 


52 


53 


2809 


148877 


7-280109 


3-75628 


53 


54 


2916 


157464 


7-348469 


3-77976 


54 


55 


3025 


166375 


7-416198 


380295 


55 


56 


3136 


175616 


7-483314 


3-82586 


56 


57 


3249 


185193 


7.549834 


3-84850 


67 


58 


3364 


195112 


7-615773 


3-87087 


58 


59 


3481 


205379 


7-681145 


3-89299 


59 


60 


3600 


216000 


7-745966 


3-91486 


60 


61 


3721 


226981 


7-810249 


3-93649 


61 


62 


3844 


238328 


7-874007 


3-95789 


62 


63 


3969 


250047 


7-937253 


3-97905 


63 


64 


4096 


262144 


8-0 


4-0 


64 


65 


4225 


274625 


8-062257 


4-02072 


65 


66 


4356 


287496 


8-124038 


4-04124 


66 


67 


4489 


300763 


8-185352 


4 06154 


67 


68 


4624 


314432 


8-246211 


4-08165 


68 


69 


4761 


328509 


8-306623 


4-10156 


69 


70 


4900 


343000 


8-366600 


412128 


70 


71 


5041 


357911 


8-426149 


4-14081 


71 


72 


5184 


373248 


8-485281 


4-16016 


72 


73 


5329 


a890l7 


8-544003 


4-17933 


73 


74 


5476 


405224 


3-602325 


4-19833 


74 


75 


5625 


421875 


8-660254 


4-21716 


75 


76 


5776 


438976 


8 717797 


4-23582 


76 


77 


5929 


456533 


8-774964 


4-25432 


77 


78 


6084 


474552 


8-831760 


4-27265 


78 


79 


6241 


493039 


8-888194 


4-29084 


79 


80 


6400 


512000 


8-944271 


4-30886 


80 


8] 


6561 


531441 


9-0 


4-32674 


81 


82 


6724 


551868 


9-055885 


4-84448 


82 


83 


6889 


571787 


9-110483 


4-36207 


88 


84 


7056 


592704 


9-165151 


4-37951 


84 


85 


7225 


614125 


9-219544 


4-39682 


85 


86 


7396 


636056 


9'27S618 


4-41400 


86 


87 


7569 


658503 


9-327379 


4-43104 


87 



16 



182 


Squares, Cubes, Square Roots, and Cube Roots. 




Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Number. 


88 


7744 


681472 


9-380831 


4 44796 


88 


89 


7921 


704969 


9-433981 


4-46474 


89 


90 


8100 


729000 


9-486833 


4-48140 


90 


91 


8281 


753571 


9539392 


4-49794 


91 


92 


8464 


778688 


9-591663 


4-51435 


92 


93 


8649 


804357 . 


9-643650 


4-53065 


93 


94 


8836 


830584 


9-695359 


4-54683 


94 


95 


9025 


857375 


9-746794 


4-56290 


95 


96 


9216 


884736 


9-797959 


4-57885 


96 


97 


9409 


912673 


9-848857 


4-59470 


97 


98 


9604 


941192 


9-899494 


4-61043 


98 


99 


9801 


970299 


9-949874 


4-62606 


99 


100 


10000 


1000000 


10-0 


4-64158 


100 


101 


10201 


1030301 


10-049875 


4-65700 


101 


102 


10404 


1061208 


10 099504 


4-67232 


102 


103 


10609 


1092727 


10-148891 


4-68754 


103 


104 


10816 


1124864 


10 198039 


4-70266 


104 


105 


11025 


1157625 


10-246950 


4-71769 


105 


106 


11236 


1191016 


10-295630 


4-73262 


106 


107 


11449 


1225043 


10-344080 


4-74745 


107 


108 


11664 


1259712 


10-392304 


4-76220 


108 


109 


11881 


1295029 


10-440306 


4-77685 


109 


110 


12100 


1331000 


10-488088 


4-79141 


110 


HI 


12321 


1367631 


10-535653 


4-80589 


111 


112 


12544 


1404928 


10-583005 


4-82028 


112 


113 


12769 


1442897 


10-630145 


4-83458 


113 


114 


12996 


] 481544 


10-677078 


4-84880 


114 


115 


13225 


1520875 


10-723805 


4-86294 


115 


116 


13456 


1560896 


10-770329 


4-87699 


116 


117 


13689 


1601613 


10-816653 


4-89097 


117 


118 


13924 


1643032 


10-862780 


4-90486 


118 


119 


14161 


1685159 


10-908712 


4-91868 


119 


120 


14400 


1728000 


10-954451 


4-93242 


120 


121 


14641 


1771561 


11-0 


494608 


121 


122 


14884 


1815848 


11-045361 


4-95967 


122 


123 


15129 


1860867 


11-090536 


4-97318 


123 


124 


15376 


1906624 


11-135528 


4-98663 


124 


125 


15625 


1953125 


11-180339 


5-0 


125 


126 


15876 


2000376 


11-224972 


5-01329 


126 


127 


16129 


2048383 


11-269427 


5-02652 


127 


128 


16384 


2097152 


11-313708 


5-03968 


128 


129 


16641 


2146689 


11-357816 


5-05277 


129 


130 


16900 


2197000 


11-401754 


5-06579 


130 


131 


17161 


2248091 


11-445523 


5-07875 


131 


132 


17424 


2299968 


11-489125 


5-09164 


132 


133 


17689 


235263*7 


11-532562 


5-10446 


133 


134. 


17956 


2406104 


11-575836 


5-11722 


134 







Squares, Cubes, Square Roots, and Cube Roots. 



183 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Number. 


135 


18225 


2460375 


11-618950 


5-12992 


135 


136 


18496 


2515456 


11-661903 


5 14256 


136 


137 


18769 


2571353 


11-704699 


5-15513 


137 


138 


19044 


2628072 


11-747340 


5-16764 


138 


139 


19321 


2685619 


11-789826 


5-18010 


139 


140 


19600 


2744000 


11-832159 


5-19249 


140 


141 


19881 


2803221 


11-874342 


5-20482 


141 


142 


20164 


2863288 


11-916375 


5-21710 


142 


143 


20449 


2924207 


11-958260 


5-22932 


143 


144 


20736 


2985984 


12-0 


5-24148 


144 


145 


21025 


3048625 


12-041594 


5-25358 


145 


146 


21316 


3112136 


12-083046 


5-26563 


146 


147 


21609 


3176523 


12-124355 


5-27763 


147 


148 


21901 


3241792 


12-165525 


5-28957 


148 


149 


22201 


3307949 


12-206555 


5-30145 


149 


150 


22500 


3375000 


12-247448 


5-31329 


150 


151 


22801 


3442951 


12-288205 


5-32507 


151 


152 


23104 


3511808 


12-328828 


5-33680 


152 


153 


23109 


3581577 


12-369316 


5-34848 


153 


154 


23716 


3652264 


12-409673 


5-36010 


154 


155 


24025 


3723875 


12-449899 


5-37168 


155 


156 


24336 


3796416 


12-489996 


5-38321 


156 


157 


24649 


3869893 


12-529964 


5-39469 


157 


158 


24964 


3944312 


12-569805 


5-40612 


158 


159 


25281 


4019679 


12-609520 


5-41750 


159 


160 


25600 


4096000 


12-649110 


5-42883 


160 


161 


25921 


4173281 


12-688577 


5-44012 


161 


162 


26244 


4251528 


12-727922 


5-45136 


162 


163 


26569 


4330747 


12-767145 


5-46255 


163 


164 


26396 


4410944 


12-806248 


547370 


164 


165 


27225 


4492125 


12-845232 


5-48480 


165 


166 


27556 


4574296 


12'884098 


5-49586 


166 


167 


27889 


4657463 


12-922848 


5-50687 


167 


168 


28224 


4^41632 


12-961481 


551784 


168 


169 


28561 


4826809 


13-0 


5-52877 


169 


170 


28900 


4913000 


13-038404 


5-53965 


170 


171 


29241 


5000211 


13-076696 


5-55049 


171 


172 


29584 


5088448 


13-114877 


5-56129 


172 


173 


29929 


5177717 


13-152946 


5-57205 


173 


174 


30276 


5268024 


13-190906 


5-58277 


174 


175 


30525 


5359375 


13-228756 


5-59344 


175 


176 


30976 


5451776 


13-266499 


5-60407 


176 


177 


31329 


5545233 


13 304134 


5-61467 


177 


178 


316S4 


5639752 


13-341664 


5-62522 


178 


179 


32041 


5735339 


13-379088 


5-63574 


179 


180 


32400 


5832000 


13-416407 


564621 


180 


181 


32761 


692)741 


13-453624 


5*65665 


181 



184 


Squares, 


Cubes, Square Roots, and Cube Roots. 




Number. 


Square. 


Cube. 


1 

j Square Root. 


Cube Root. 


Number. 


182 


33124 


6028568 


13-490737 


5-66705 


182 


183 


33489 


6128487 


13-527749 


5-67741 


183 


184: 


33856 


6229504 


13-564660 


5-68773 


184 


185 


34225 


6331625 


13-601470 


5-69801 


185 


186 


34596 


6434856 


13-638181 


5-70826 


186 


187 


34969 


6539203 


13-674794 


5-71847 


187 


188 


35344 


6644672 


13-711309 


5-72865 


188 


189 


35721 


6751269 


13-747727 


5-73879 


189 


190 


36100 


6859000 


13-784048 


5-74889 


190 


191 


36481 


6967871 


13-820275 


5-75896 


191 


192 


36864 


7077888 


13-856406 


5-76899 


192 


193 


37249 


7189057 


13-892444 


5-77899 


193 


194 


37636 


7301384 


13-928388 


5-78896 


194 


195 


38025 


7414875 


13-964240 


5-79889 


195 


196 


38416 


7529536 


140 


5-80878 


196 


197 


38809 


7645373 


14-035668 


5-81864 


197 


198 


39204 


7762392 


14-071247 


5-82847 


198 


199 


39601 


7880599 


14-106736 


5-83827 


199 


200 


40000 


8000000 


14-142135 


5-84803 


200 


201 


40401 


8120601 


14-177446 


5-85776 


201 


202 


40804 


8242408 


14-212670 


5-86746 


202 


203 


41209 


8365427 


14-247806 


5-87713 


203 


204 


41616 


8489664 


1428-2856 


5-88676 


204 


206 


42025 


8615125 


14-317821 


5-89636 


205 


206 


42436 


8741816 


14-352700 


6-90594 


206 


207 


42849 


8869743 


14-387494 


5-91548 


207 


208 . 


43264 


8998912 


14-422205 


5-92499 


208 


209 


43681 


9129329 


14-456832 


6-93447 


209 


210 


44100 


9261000- 


14-491376 


5-94392 


210 


211 


44521 


9393931 


14-525839 


5-95334 


211 


212 


44944 


9528128 


14-560219 


5-96273 


212 


213 


45369 


9663597 


14-594519 


5-97209 


213 


214 


45796 


9800344 


14-628738 


5-98142 


214 


215 


46225 


9938375 


14-662878 


5-99072 


215 


216 


46656 


10077696 


14-696938 


6-0 


216 


217 


47089 


10218313 


14-730919 


6-00924 


217 


218 


47524 


10360232 


14-764823 


6-01846 


218 


219 


47961 


10503459 


14-798648 


6-02765 


219 


220 


48400 


10648000 


14-832397 


6-03681 


220 


221 


48841 


10793861 


14-866068 


6-04594 


221 


^2 


49284 


10941048 


14-899664 


6-05504 


222 


223 


49729 


11089567 


14-9331.84 


6-06412 


223 


224 


50176 


11239424 


14-966629 


6-07317 


224 


225 


50625 


11390625 


15-0 


6-08220 


225 


226 1 


51076 


11543176 


15-033296 


6 09119 


226 


227 j 


51529 


11697083 


15-066519 


6-10017 


227 


228 1 


51984 


11852352 


15-099668 


6-10911 


228 











Squares, 


Cubes, Square Roots, and Cube Roots. 


185 












Number. 


Number 


( 

Square. 


Cube. 


Square Root. 


Cube Root. 




52441 


12008989 


15-132746 






229 


6-11803 


229 


280 


52900 


12167000 


15-165750 


6-12692 


230 


231 


53361 


12326391 


15-198684 


6-13579 


231 


232 


53824 


12487168 


15-231546 


6-14463 


232 


233 


54289 


12649337 


15-264337 


6-15344 


233 


234 


54756 


12812904 


15-297058 


616224 


234 


235 


55225 


12977875 


15-329709 


6-17100 


235 


236 


55696 


13144256 


15-362291 


6-17974. 


236 


237 


56169 


13312053 


15-394804 


6-18846 


287 


238 


56644 


13481272 


15-427248 


6-19715 


238 


239 


57121 


13651919 


15-459624 


6-20582 


289 


240 


57600 


18824000 


15-491933 


6-21446 


240 


241 


58081 


13997521 


15-624174 


6-22308 


241 


242 


58564 


14172488 


15-556349 


6 23167 


242 


243 


59049 


14348907 


15-588457 


0^4025 


243 


244 


59536 


14526784 


15-620499 


6-24879 


244 


245 


60025 


34706125 


15-652475 


6-25732 


245 


246 


60516 


14886936 


15-684387 


6-26582 


246 


247 


61009 


15069223 


15-716233 


6-27430 


247 


248 


61504 


15252992 


15-748015 


6-28276 


248 


249 


62001 


15438249 


15-779733 


6-29139 


249 


250 


62500 


15625000 


15-811388 


6-29960 


250 


251 


63001 


15813251 


15-842979 


6-30799 


251 


252 


63504 


16003008 


15-874507 


6-31635 


252 


253 


64009 


16194277 


15-905973 


6-32470 


253 


254 


64516 


16387064 


15-937377 


6-33302 


254 


255 


65025 


16581375 


15-908719 


6-34132 


255 


256 


65536 


16777216 


16- 


6-34960 


256 


257 


66049 


3 6974593 


16-03123 9 


6-35786 


257 


258 


66564 


17173512 


36-062378 


6-36609 


258 


259 


67081 


17373979 


16 093476 


637431 


259 i 


260 


67600 


17576000 


16 124515 


6-38250 


260 


261 


68121 


17779581 


16-155494 


6-39067 


261 


262 


68644 


17984728 


16-186414 


6-39882 


262 


263 


69169 


18191447 


16-217274 


6-40695 


263 


264 


69696 


18399744 


16-248076 


6-43506 


264 


265 


70225 


18609625 


16-278820 


6 42315 


265 


266 


70756 


18821096 


3 6-309506 


648322 


266 


267 


71289 


19034163 


16-3403 34 


6-43927 


267 


268 


71824 


3 9248832 


3 6-370705 


6-44730 


268 


269 


72361 


19465109 


16-401219 


C-45531 


269 


270 


72900 


19683000 


3 6-431076 


6 46830 


270 


271 


73441 


19902511 


10-462077 


6-47127 


271 


272 


78984 


20123648 


16-492422 


6-47922 


272 


273 


74529 


20346417 


16522731 


48715 


273 


274 


75076 


20570824 


3 0-552945 


6-^^9506 


274 


275 


75625 


20796875 


16-583124 


6-50295 


275 

i 



16* 



186 


Squares, 


Cubes, Square Roots, and Cube Roots. 




Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Number. 








276 


76176 


21024576 


16-613247 


6-51083 


276 


277 


76729 


21253933 


16-643317 


6-51868 


277 


278 


77284 


21484952 


16673332 


6-52651 


278 


279 


77841 


21717639 


16-703293 


6-53433 


279 


280 


78400 


21952000 


16-733200 


6-54213 


280 


281 


78961 


22188041 


16-763054 


6-54991 


281 


282 


79524 


22425768 


16-792855 


6-55767 


282 


283 


80089 


22665187 


16-822603 


6-56541 


283 


284 


80656 


22906304 


16-852299 


6-57313 


284 


285 


81225 


23149125 


16-881943 


6-58084 


285 


286 


81796 


23393656 


16-911534 


6-58853 


286 


287 


82369 


23639903 


16-941074 


6-59620 


287 


288 


82944 


23887872 


16-970562 


6-60385 


288 


289 


83521 


24137569 


17 


6-61148 


289 


290 


84100 


24389000 


17-029386 


6-61910 


290 


291. 


84681 


24642171 


17-058722 


6-62670 


291 


292 


85264 


24897088 


17-088007 


6-63428 


292 


293 


85849 


25153757 


17-117242 


6-64185 


293 


294 


86436 


25412184 


17-146428 


6-64939 


294 


295 


87025 ; 


25672375 


17-175564 


6-65693 


295 


296 


87616 


25934336 


17-204650 


6-66444 


296 


297 


88209 


26198073 


17-233687 


6-67194 


297 


298 


88804 


26463592 


17-262676 


6-67942 


298 


299 


89401 ! 


26730899 


17-291616 


6-68688 


299 


300 


90000 i 


27000000 


17-320508 


6-69432 


300 


301 


90601 i 


27270901 


17 349351 


6-70175 


301 


302 


91204 i 


27543608 


17-378147 


6-70917 


302 


303 


91809 : 


27818127 


17-406895 


6^71657 


303 


304 


92416 


28094464 


17-435595 


6-72395 


304 


305 


93025 


28372625 


17-464249 


6-73131 


305 


306 


93636 


28652616 


17-492855 


6-73866 


306 


307 


94249 


28934443 


17-521415 


6-74599 


307 


308 


94864 


29218112 


17-549928 


6-75331 


308 


309 


95481 


29503629 


17-578395 


6-76061 


309 


310 


96100 


29791000 


17-606816 


6-76789 


310 


311 


96721 


30080231 


17-635192 


6-77516 


311 


312 


97344 


30371328 


17-663521 


6-78242 


312 


313 


97969 


30664297 


17-691806 


6-78966 


313 


314 


98596 


30959144 


17-720045 


6-79688 


314 


315 


99225 


31255875 


17-748239 


6-80409 


315 


316 


99856 


31554496 


17-776388 


6-81128 


316 


317 


100489 


31855013 


17 804493 


6-81846 


317 


318 


i 101124 


32157432 


17-832554 


6-82562 


318 


319 


1 101761 


32461759 


17860571 


6-83277 


319 


320 


' 102400 


32768000 


17-888543 


6-83990 


320 


321 


103041 


33076161 


17-916472 


6-84702 


321 


322 


103684 


33386248 


17-944358 


6-85412 


322 















Squares, 


Cubes, Square 


Roots, and Cube Roots. 


187 


Number. 


Square. 


Cube. 


Square Root. 


Cube Root 


Number. 


323 


104329 


33698267 


17-972200 


6-86121 


323 


324 


104976 


84012224 


18-0 


6-86828 


324 


325 


105625 


34328125 


18-027756 


6 87584 


325 


326 


106276 


34645976 


18-055470 


6-88238 


826 


327 


106929 


34965783 


18-083141 


6-88941 


327 


328 


107584 


35287552 


18-110770 


6-89643 


328 


329 


108241 


35611289 


18-188357 


6-90343 


329 


330 


108900 


35937000 


18-165902 


6-91042 


380 


331 


109561 


36264691 


18193405 


6-91739 


331 


332 


110224 


36594368 


18-220867 


6-92435 


382 


333 


110889 


36926037 


18-248287 


6-93180 


333 


334 


111556 


37259704 


18-275666 


6-93823 


334 


335 


112225 


37595375 


18-303005 


6-94514 


335 


336 


112896 


37933056 


18-330802 


6-95205 


336 


337 


113569 


38272753 


18357559 


6-95894 


837 


338 


114244 


38614472 


18-384776 


6-96581 


388 


339 


114921 


38958219 


18-411952 


6 97268 


339 


340 


115600 


39304000 


18-489088 


6-97953 


340 


341 


116281 


39651821 


18-466185 


6-98636 


341 


342 


116964 


40001688 


18-493242 


6-99319 


342 


343 


117649 


40353607 


18-520259 


7-0 


343 


344 


118336 


40707584 


18-547237 


7-00679 


344 


345 


119025 


41063625 


18-574175 


7-01357 


345 


346 


119716 


41421736 


18-601075 


7-02034 


346 


347 


120409 


41781923 


18-627936 


7-02710 


847 


348 


121104 


42144192 


18-654758 


7-03384 


348 


349 


121801 


42508549 


18-681541 


7-04058 


349 


350 


122500 


42875000 


18-708286 


7-04729 


350 


351 


123201 


43243551 


18-784994 


7-05400 


851 


352 


123904 


43614208 


18-761663 


7-06069 


352 


353 


124609 


43986977 


18-788294 


7-06737 


353 


354 


125316 


44361864 


18-814887 


7-07404 


854 


355 


126025 


44738875 


18-841443 


7 08069 


355 


356 


126736 


45118016 


18-867962 


7-08734 


356 


857 


127449 


45499293 


18-894443 


7-09397 


357 


358 


128164 


45882712 


18-920887 


7-10058 


358 


359 


128881 


46268279 


18-947295 


7-10719 


359 


360 


129600 


46656000 


18-973666 


7-11878 


860 


361 


130321 


47045881 


19-0 


7-12036 


361 


362 


131044 


47437928 


19-026297 


7-12693 


862 


363 


131769 


47832147 


19-052558 


7-13849 


363 


864 


132496 


48228544 


19-078784 


7-14008 


364 


865 


133225 


48627125 


19-104973 


7-14656 


365 


366 


133956 


49027896 


19-131126 


7-15309 


366 


367 


134689 


49430863 


19-157244 


7-15959 


367 


368 


135424 


49886032 


19-183326 


7-16609 


368 


369 


136161 


50243409 


19-209372 


7-17258 


869 





188 


Squares 


Cubes, Square Roots, and ( 


OuDE Roots. 






Number 


. I Square. 

1 


Cube. 


Square Root, 


Cube Root. 


Number. 


370 


1 136900 


50653000 


19-235384 


7-17905 


370 




371 


i 137641 


51064811 


19-261360 


7-18551 


371 




372 


138384 


51478848 


19-287301 


7-19196 


372 




373 


139129 


51895117 


19-813207 


7-19840 


373 




374 


139876 


52313624 


19-839079 


7-20483 


374 




375 


140625 


52734375 


19-364916 


7-21124 


375 




376 


141376 


63157376 


19 890719 


7-21765 


376 




377 


142129 


53582683 


19-416487 


7-22404 


377 




378 


142884 


54010152 


19-442222 


7-23042 


378 




379 


143641 


54439939 


19-467922 


7-23679 


379 




380 


144400 


54872000 


19-493588 


7-24315 


380 




381 


145161 


55306841 


19-519221 


7-24950 


881 




382 


145924 


55742968 


19-544820 


7-25584 


882 




383 


146689 


56181887 


19-670385 


7-26216 


383 




384 


147456 


56628104 


19 595917 


7-26848 


884 




385 


148225 


57066625 


19-621416 


7 27478 


385 




386 


148996 


57512456 


19-646882 


7-28107 


386 




387 


149769 


67960603 


19-672815 


7-28786 


387 




388 


150544 


5-^411072 


19-697715 


7-29863 


388 




389 


151821 


58863859 


19-728082 


7-29989 


389 




390 


152100 


59819000 


19-748417 


7-30614 


390 




391 


152881 


59776471 


19-778719 


7-21238 


391 




392 


153664 


60286288 


19-798989 


7-31861 


392 




398 


154449 


60698457 


19-824227 


7-32482 


393 




394 


155236 


61162984 


19-849438 


7-33103 


394 




895 


156025 


61629875 


19-874606 


7-83723 


395 




896 


156816 


62099186 


19-899748 


7-34342 


396 




397 


157609 


62570773 


19-924858 


7-84959 


397 




398 


158404 


63044792 


19949937 


7-35576 


398 




399 


159201 


63521199 


19-974984 


7-36191 


399 




400 


160000 


64000000 


20-0 


7-36806 


400 




401 


160801 


64481201 


20-024984 


7-37419 


401 




402 


161604 


64964808 


20-049987 


7-38032 


402 




403 


162409 


65450827 


20-074859 


7-38643 


403 




404 1 


163216 


65939264 | 


20-099751 


7-39254 


404 




405 


164025 


66430125 


20-124611 


7-39863 


405 




406 


164836 


66928416 


20-149441 


7-40472 


406 




407 


165649 


67419148 


20-174241 


7-41079 


407 




408 i 


166464 


67917812 


29-199009 


7-41685 


408 




409 


167281 


68417929 


20-223748 


7-42291 


409 




410 


168100 


68921000 


20-248456 


7-42895 


410 




411 


168921 


69426581 


20-273134 


7-43499 


411 




412 


169744 


69934528 


20-297788 


7-44101 


412 




413 


170569 


70444997 


20-322401 


7-44703 


413 




414 


171396 


70957944 


20-346989 


7-45303 


414 




415 


172225 


71473375 


20-371548 


7-45903 


415 




416 


173056 


71991296 


20-396078 


7-46502 


416 





Squares, Cubes, Square Roots, and Cube Roots. 


189 


Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Nuniber. 






417 


1 — 

173889 


72511713 


20-420577 


7-47099 


417 i 


418 


174724 


73034632 


20-445048 


7-47696 


418 


419 


175561 


73560059 


20-469489 


7-48292 


419 


420 


176400 


74088000 


20-493901 


7-48887 


420 


421 


177241 


74618461 


20-518284 


7-49481 


421 


422 


178084 


75151448 


20-542638 


7-50074 


422 


423 


178929 1 


75686967 


20-566963 


7-50666 


423 


424 


179776 


76225024 


20-591260 


7-51257 


424 


425 


180625 


76765625 


20-615528 


7-51847 


425 


426 


181476 


77308776 


20-639767 


7-52436 


426 


427 


182329 


77854483 


20-663978 


7-53024 


427 


428 


183184 


78402752 


20-688160 


7-53612 


428 


429 


184041 


78953589 


20-712315 


7-54198 


429 


430. 


184900 


79507000 


20-736441 


7-54784 


430 


431 


185761 


80062991 


20-760539 


7-55368 


431 


432 


186624 


80621568 


20-784609 


7-55952 


432 


433 


187489 


81182737 


20-808652 


7-56535 


433 


434 


188356 


81746504 


20-832666 


7-57117 


434 


435 


189225 


82312875 


20-856653 


7 57698 


435 


436 


190096 


82881856 


20-880613 


7-58278 


436 


437 


190969 


83453453 


20-904545 


7-58857 


437 


438 


191844 


84027672 


20-928449 


7-59436 


438 


439 


192721 


84604519 


20-952326 


7-60013 


439 


! 440 


193600 


85184000 


20-976177 


7-60590 


440 


} 441 


194481 


85766121 


21-0 


7-61166 


441 


1 442 


195364 


86350888 


21-023796 


7-61741 


442 


j 443 


196249 


86938307 


21-047565 


7-62315 


443 


444 


197136 


87528384 


21-071307 


7-62888 


444 


445 


198025 


88121125 


21-095023 


7-63460 


445 


446 


198916 


88716536 


21-118712 


7-64032 


446 


447 


199809 


89314623 


21-142374 


7-64602 


447 


448 


200704 


89915392 


21-166010 


7-65172 


448 


449 


201601 


90518849 


21-189620 


7.65741 


449 


450 


202500 


91125000 


21-213203 


7-66309 


450 


451 


203401 


91733851 


21-236760 


7-66876 


451 


452 


204304 


92345408 


21-260291 


7-67443 


452 


453 


205209 


92959677 


21-283796 


7-68008 


453 


454 


206116 1 


93576664 


21-307275 


7-68573 


454 


455 


207025 1 


94196375 


21-330729 


7-69137 


455 


456 


207936 


94818816 


21-354156 


7-69700 


456 


457 


208849 


95443993 


21-377558 


7-70262 


457 


458 


209764 


96071912 


21-400934 


7-70823 


458 


459 


210681 


96702579 


21-424285 


7-71384 


459 


460 


211600 


97336000 


21-417610 


7-71944 


460 


461 


212521 


97972181 


21-470910 


7-72503 


461 


462 


213444 


98611128 


21-494185 


7-73061 


462 


463 


214369 


99252847 


21-517434 


7-73618 


463 



IQO 



Squares, Cubes, Square Roots, axd Cube Roots, 



Number 


! 

1 Square. 


i 

i Cube. 


Square Root. 


■ Cube Root. 

i 


Number. 


464 


215296 


99897344 


21-540659 


7-74175 


464 


465 


2]6225 


100544625 


21-563858 


7-74731 


465 


466 


217156 


101194696 


21-587033 


7-75286 


466 


467 


218089 


101847563 


21-610182 


7-75840 


467 


468 


219024 


102503232 


21-633307 


7-76393 


468 


469 


219961 


103161709 


21-656407 


7-76946 


469 


470 


220900 


103S23000 


21-679483 


7-77498 


470 


471 


221841 


101487111 


21-702534 


7-78049 


471 


472 


222784 


105154048 


21-725561 


7 78599 


472 


473 


223729 


105823817 


21-748563 


7-79148 


473 


474 


224676 


106496424 


i 21-771541 


7-79697 


474 


475 


225625 


107171875 


21-794494 


7-80245 


475 


476 


226576 


107850176 


21-817424 


7-80792 


476 


477 


227529 


108531333 


21-840329 


7-81338 


477 


478 


228484 


109215352 


21-863211 


7-81884 


478 


479 


229441 


109902239 


21-886068 


7-82429 


479 


480 


230400 


110592000 


21-908902 


7-82973 


480 


481 


231361 


111284641 


21-931712 


7-83516 


481 


482 


232324 


111980168 


21-954498 


7-84059 


482 


483 


233289 


112678587 


21-977261 


7-84601 


483 


484 


234256 


113379904 


22-0 


7-85142 


484 


485 


235225 


114084125 


22-022715 


7-85682 


485 


486 


236196 


114791256 


22-045407 


7-86222 


486 


487 


237169 


115501303 


22 068076 


7-86761 


487 


488 


238144 


116214272 


22090722 


7-87299 


488 


489 


239121 


116930169 


22-113344 


7-87836 


489 


490 


240100 


117649000 


22-135943 


7-88378 


490 


491 


241081 


118370771 


22-158519 


7-88909 


491 


492 


242064 


119095488 


22-181073 


7-89444 


492 


493 


243049 


119823157 


22-203603 


7-89979 


493 


494 


244036 


120553784 


22-226110 


7 90512 


494 


495 


245025 


121287375 


22-248595 


7-91045 


495 


496 


246016 


122023936 


22-271057 


7-91578 


496 


497 


247009 


122763473 


22-293496 


7-92109 


497 


498 


248004 


123505992 


22-315913 


7-92640 


498 


499 


249001 


124251499 


22-338307 


7-93171 


499 


500 


250000 


1 25000000 


22 360679 


7-93700 


500 


501 


251001 


125751501 


22-383029 


7-94229 


501 


502 


252004 


126506008 


22-405356 


7-94757 


502 


503 


253009 


127263527 


22-427661 


7-95284 


503 


504 


254016 


128024064 


22-449944 


795811 


504 


505 


255025 


128787625 


22-472205 


7-96337 


505 


506 


256036 


129554216 


22-494443 


7-96862 


506 


oOl 


257049 1 


130323843 


22-516660 


7-97387 


507 


508 


258064 1 


131096512 


22-538855 


7-97911 


508 


509 


259081 


131872229 


22-561028 


7-98434 


509 


510 


260100 

1 


132651000 


22-583179 


7-98956 


510 





Squares, Cubes, Square Roots, and Cube Roots. 


191 


Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Number. 








511 


261121 


133432831 


22-605309 


7-99478 


511 


512 


262144 


134217728 


22-627417 


8-0 


512 


513 


263169 


135005697 


22-649503 


8-00520 


518 


514 


264196 


135796744 


22-671568 


8-01040 


514 


515 


265225 


136590875 


22-693611 


8-01559 


515 


516 


266256 


137388096 


22-715633 


8-02077 


516 


517 


267289 


138188413 


22-737634 


8-02595 


517 


518 


268324 


138991832 


22-759613 


8-03112 


518 


519 


269361 


139798359 


22-781671 


8-03629 


519 


520 


270400 


140608000 


22-803508 


8-04145 


520 


521 


271441 


141420761 


22-825424 


8-04660 


521 


522 


272484 


142236648 


22-847319 


8-05174 


522 


523 


273529 


143055667 


22-869193 


8-05688 


523 


524 


274576 


143877824 


22-891046 


806201 


524 


525 


275625 


144703125 


22-912878 


8-06714 


525 


526 


276676 


145531576 


22-934689 


8-07226 


526 


527 


277729 


146363183 


22-956480 


8-07737 


527 


528 


278784 


147197952 


22-978250 


8-08248 


528 


529 


279841 


148035889 


23-0 


8-08757 


529 


530 


280900 


148877000 


23-021728 


8-09267 


530 


531 


281961 


149721291 


23-043437 


8-09775 


531 


532 


283024 


150568768 


23-065125 


810283 


532 


533 


284089 


151419437 


23-086792 


8-10791 


533 


534 


285156 


152273304 


23-108440 


8-11298 


534 


535 


286225 


153130375 


23-130067 


8-11804 


535 


536 


287296 


153990656 


23-151673 


8-12309 


536 


537 


288369 


154854153 


23-173260 


8-12814 


537 


538 


289444 


155720872 


23-194827 


8-13318 


538 


539 


290521 


156590819 


23-216373 


8-13822 


539 


540 


291600 


157464000 


23-237900 


8-14325 


540 


541 


292681 


158340421 


23-259406 


8-14827 


541 


542 


293764 


159220088 


23-280893 


8-15329 


542 


543 


294849 


160103007 


23-302360 


3-15830 


543 


544 


295936 


160989184 


23-323807 


8-16331 


544 


545 


297025 


161878625 


23 345235 


816830 


545 


546 


298116 


162771386 


23-366642 


8-17330 


546 


547 


299209 


163667323 


23-388031 


8-17828 


547 


548 


300304 


164566692 


23-409399 


8-18326 


548 


549 


801401 


165969149 


23-430749 


8-18824 


649 


550 


302500 


166375000 


23-452078 


8-19321 


550 


551 


303601 


167284151 


23-473389 


8-19817 


551 


562 


304704 


168196608 


23-494680 


8-20313 


552 


653 


305809 


160112377 


23-515952 


8-2U8U8 


553 


654 


806916 


170031464 


23-537204 


8-21302 


554 


655 


308025 


170953875 


23-558438 


8-21796 


565 


666 


809136 


171879616 


23-579652 


8-22289 


556 


657 


310249 


172808693 


23-600847 


8-22782 


657 













192 


Squares, 


Cubes, Square Roots, and Cube Koots, 




Kumber 


Square 


Cube 


Square Root: 


Cube Root. 


Number. \ 

i 


658 


311364 


173741112 


23-622023 


8-23274 


558 


559 


312481 


174676879 


23-643180 


8-23766 


559 


560 


313600 


175616000 


23-664319 


8-24257 


560 1 


561 


314721 


176658481 


23-685438 


8-24747 


561 ! 


562 


315844 


177604328 


23-706539 


8-25237 


562 j 


563 


316969 


178453547 


23-727621 


8-25726 


563 [ 


564 


318096 


179406144 


23-748684 


8-26214 


564 [ 


565 


319225 


180362125 


23-769728 i 


8-26702 


565 i 


566 


320356 


181321496 


23-790754 i 


8-27190 


566 ; 


567 


321489 


182284263 


23-811761 1 


8-27677 


567 1 


568 


322624 


183250432 


23-832760 


8-28163 


568 1 


569 


823761 


184220009 


23-853720 


8-28649 


569 ! 


570 


324900 


185193000 


23-874672 


8-29134 


570 


571 


326041 


186169411 


23-895606 


8-29619 


571 


572 


327184 


187149248 


23-916521 


8-30103 


572 1 


573 


328329 


188132517 


23-937418 


8-30586 


573 i 


574 


329476 


189119224 


23-958297 


8-31069 


574 


575 


330626 


190109375 


23-979157 


8-31551 


575 


576 


331776 


191102976 


24-0 


8-32033 


576 


577 


332929 


192100033 


24-020824 


8-32514 


577 


578 


334084 


193100552 


24-041630 


8-32995 


578 


579 


336241 


194104539 


24-062418 


8-33475 


579 


580 


336400 


196112000 


24-083189 


8-33965 


580 


581 


337661 


196122941 


24-103941 


8-34434 


681 


582 


338724 


197137368 


24-124676 


834912 


582 


583 


339889 


198156287 


24-145392 


8-35390 


583 


584 


341056 


199176704 


24-166091 


8 '36867 


584 


585 


342225 


200201625 


24-186773 


8-36344 


585 


586 


343396 


201230066 


24-207436 


8-36820 


586 


587 


344569 


202262003 


24-228082 


8-37296 


587 


588 


345744 


203297472 


24-248711 


8-37771 


588 


589 


346921 


204336469 


24-269322 


8-38246 


589 


590 


348100 


205379000 


24-289915 


8-38720 


590 


691 


349281 


206425071 


24-310491 


8-39194 


591 


592 


350464 


207474688 


24-331060 


8-39667 


592 


593 


351649 


208527857 


24-861591 


8-40139 


593 i 


594 


352836 


209584584 


24-372115 


8-40611 


694 : 


595 


354025 


210644875 


24-392621 


8-41083 


595 , 


596 


365216 


211708736 


24-413111 i 


8-41554 


596 


597 


356409 


212776173 


24-433583 


8-42024 


&97 


598 


357604 


213847192 


24-464038 


8-42494 


598 


599 


358801 


214921799 


24-474476 


8-42963 


599 


600 


360000 


216000000 


24-494897 


8-43432 


600 


601 


361201 


217081801 


24-616801 


8-43900 


601 


602 


362404 


218167208 


24-536688 


8-44368 


602 


603 


363609 


219256227 


24-556058 


8-44836 


60^ 


604 


364816 


220348864 


24-576411 


8 -45^02 


604 















Squares, Cubes, Square Roots, and Cube Roots. 



19.S 





Square. 


Cube. 








Number. 


Sfiuare Root. 


Cube Root. 


Number. 


605 


36r3025 


221445125 


24-596747 


8-45769 


605 


606 


867236 


222545016 


24-617067 


8-46234 


606 


607 


368449 


223648543 


24-637370 


8-46700 


607 


608 


369664 


224755712 


24-657656 


8-47164 


608 


609 


370881 


225866529 


24-677925 


8-47628 


609 


610 


372100 


226981000 


24-698178 


8-48092 


610 


611 


373321 


228099131 


24-718414 


8-48555 


611 


612 


374544 


229220928 


24-738633 


8-49018 


612 


613 


375769 


230346397 


24-758836 


8-49480 


613 


614 


876996 


281475544 


24-779023 


8-49942 


614 


615 


878225 


232608375 


24-799193 


8-50403 


615 


616 


379456 


233744896 


24-819847 


8 50864 


616 


617 


380689 


234885113 


24-839484 


8-51324 


617 


618 


381924 


236029032 


24-859605 


8-51784 


618 


619 


883161 


237176659 


24-879710 


8-52243 


619 


620 


384400 


238328000 


24-899799 


8-52701 


620 


621 


385641 


239483061 


24-919871 


853160 


621 


622 


886884 


240641848 


24-939927 


8-53617 


622 


623 


388129 


241804367 


24-959967 


8-54075 


623 


624 


889876 


242970624 


24-979992 


8-54531 


624 


625 


390625 


244140625 


25-0 


8-54987 


625 


626 


391876 


245814376 


25-019992 


8 55443 


626 


627 


398129 


246491883 


25-039968 


8 55898 


627 


628 


394384 


247673152 


25-059928 


8-56353 


628 


629 


395641 


248858189 


25-079872 


8-56808 


629 


630 


396900 


250047000 


25-099800 


8-57261 


630 


631 


898161 


251289591 


25-119713 


8-57715 


631 


632 


399424 


252435968 


25-189610 


8-58168 


632 


633 


400689 


258686137 


25159491 


8 58620 


633 


634 


401956 


254840104 


25179856 


8-59072 


634 


635 


403225 


256047875 


25-199206 


8-59523 


635 


636 


404496 


257259456 


25219040 


8-59974 


636 


637 


405769 


258474853 


25-238858 


8-60425 


637 


638 


407044 


259694072 


25-258661 


8-60875 


638 


639 


408321 


260917119 


25-278449 


8-61324 


639 


640 


409600 


262144000 


25-298221 


8-61773 


640 


641 


410881 


263374721 


25-817977 


8-62222 


641 


642 


412164 


264609288 


25-887718 


8-62670 


642 


643 


413449 


265847707 


25 857444 


8'63118 


643 


644 


414736 


267089984 


25-877155 


8-68565 


644 


645 


416025 


268336125 


25-896850 


8-64012 


645 


646 


417316 


269586136 


25-416530 


8-64458 


646 


647 


418609 


270840023 


25-436194 


8-64904 


647 


648 


419904 


272097792 


25-455844 


8-65349 


648 


649 


421201 


278859449 


25-475478 


8 65794 


649 


650 


422500 


274625000 


25-495097 


8 66239 


650 


651 


423801 


275894451 


25-514701 


8-66683 


651 



17 



194 



S-o-'AP.i:-. Crsisi, Square Pv.x*!^ and Ccbe Root?. 



i~ 




C u r«? 


S-Mire Root- 


CuS? Roo:- 


y-izuz^r. 


^ 65. 


425104 


2T71r37S>'S 


25 534290 


8-67126 


652 


653 


426409 


27S4450T7 


25-5538fr4 


8-67569 


653 


651 


427716 


279726264 


, 25-573423 


8-68012 


654 


655 


429025 


2S1011375 


25-592967 


8-68454 


655 


656 


430336 


282300416 


' 25-612496 


8-68896 


636 


65T 


431649 


283593393 


25 632011 


8-69337 


^ 


658 


432964 


2S4S9<jS12 


25-651510 


8-69778 


^a 


659 


434281 


2S6191179 


; 25-670995 


; 8^0218 


^i 


660 


435600 


2S7496000 


: 25-690465 


8^0658 


m 


661 


436921 


2SS8Cm:781 


; 25-709920 


; 8^1098 


m 


662 


438244 


290117528 


25^29360 


; 8-71537 


' 662 


663 


439569 


291434247 


' 25^48786 


i 8^1975 


663 1 


664 


440896 


292754944 


; 25-768197 


. 8^2414 


^ 664 1 


665 


442225 


294079625 


25^87593 


stl^-: 


665 J 


666 


443556 


2954C)S296 


25 806975 


s-->-; 


666 1 


66T 


444^89 


29674«:i963 


2-'> ^L'":'-i;~ 


8'7S726 


^ 


66S 


446224 


298077632 


2-'''^-': -' - ■"■ 


8^4162 


669 


447561 


299418309 


25-5 65';'£.4 


8-74^^^ 


o69 


670 


448900 


300763000 


25-884358 


b ■" " " - 


o70 


6V1 


450241 


302111711 


25-903667 


: i':^ 


i-Tl 


672 


451584 


303464448 


25-922962 


■ 6-: 


e72 


673 


452929 


S.:4S21217 


25-942243 


■ S 7^ - 


':■"? 


674 


454276 


3r»6182024 


25-961510 


i ^ 


■ "- 


675 


455625 


307546875 


25-980762 


: s---- 


- - ^ 


676 


455976 


30S915776 


26-0 


- s-:;. 


■ - 


67V 


45S329 


3102SS733 
311665752 


26D19223 


8-7. -: 


. 7 7 


678 


4596S4 


26-038433 


" s--:. 


o78 


679 


461041 


313046839 


26-057628 


; s-7-;. 


r>79 


680 


462400 


314432000 


26-076809 


87 


-:'?''' 


681 


463761 


315821241 


26-095976 


8-:- . 


^51 


682 


465124 


31721456S 


-26-115129 


8-s;l.- 


-: xw 


683 


466489 


31S61I9S7 


26 134268 


8-S;:.:- 


'■ ^ ;; 


684 


467856 


32«Xj13504 , 


26-153393 


: 8-si':.-; 


: .. 


685 


469225 


321419125 : 


26172504 


5 8 8151: 


: -: 


686 ' 


470596 


32i'?'2^^-' 


26-191601 


8-4. 




687 


471969 


3iI:I^- 


26-210684 


8i_ 




688 


473344 


3icf,:>7i. . 


26-229754 


8^ 


^v 


689 


474721 


32:nS2769 [ 


26-248S09 


8S ... 


':S9 


690 , 


4761CK) 


S2S509OOO t 


26-267851 


8- - 


'^90 


691 


477481 


329939371 1 


26-286878 : 


8-^. 1 


991 


692 r 


478864 


SSI373888 


26-305892 


8S. V. 


392 


693 


480249 


332S12557 I 


26-3-24893 ; 


SS >4 


o93 


eu 


481636 


334255384 


26-343879 ; 


8S- : 


o94 


695 


4S3025 


335702375 ' 


26-362852 l 


8-857 S4 


695 


696 


4S4416 


337153536 


26-381811 


8-862f^'rf 


696 


69t 


485809 


338608873 


26-400757 J 


8-s-^:^; 


r97 


698 


487204 


^40068392 * 


26-419689 ^ 


8-5- - 


e9S 



Squares, Cubes, Square Roots, and Cube Roots. 



195 



1 

N umber. 


Square. 


Cube. 








Square Root. 


Cube Root. 


Number. 


699 


488601 


341532099 


! 26-438608 


8-87480 


699 


700 


490000 


343000000 


26-457513 


8-87904 


700 


701 


491401 


344472101 


26-476404 


8-88326 


701 


702 


492804 


345948408 


26-495282 


8-88748 


702 


703 


494209 


347428927 


26-514147 


8-89170 


703 


704 


495616 


348913664 


26-532998 


8-89592 


704 


705 


497025 


350402625 


26-551836 


8-90013 


705 


705 


498436 


351895816 


26-570660 


8^90433 


706 1 


707 


499849 


353393243 


26-589471 ,, 


i 8-90853 


707 


708 


501264 


354894912 


26-608269 


8-91273 


708 


709 


502681 


356400829 


26-627053 


8-91693 


709 


710 


504100 


357911000 


26-645825 


8-92112 


710 


711 


505521 


359425431 


26-664583 


8-92530 


711 


712 


507944 


360944128 


26-683328 


8-92949 


712 


713 


508369 


362467097 


26-702059 


^ 893366 


713 


714 


509796 


363994344 


26-720778 


8-93784 


714 


715 


511225 


365525875 


26-739483 


8-94201 


715 


716 


512656 


367061696 


26-758176 


8-94618 


716 


717 


514089 


368601813 


26-776855 


8-95034 


717 


718 


515524 


370136232 


26-795522 


8-95450 


718 


719 


516961 


371694959 


26-814175 


8-95865 


719 


720 


518400 


373248000 


26-832815 


8-96280 


720 


721 


519841 


374805361 


26-851443 


8-96695 


721 


722 


521284 


376367048 


26-870057 


8-97110 


722 


723 


522729 


377933067 


26-888659 


8-97524 


723 


724 


524176 


379503424 


26-907248 


8-97937 


724 


725 


525625 


381078125 


26-925824 


8-98350 


725 


726 


527076 


382657176 


26-944387 


8-98763 


726 


727 


528529 


384240583 


26-962937 


8-99176 


727 


728 


529984 


385828352 


26-981475 


8-99588 


728 


729 


531441 


387420489 


27-0 


9-0 


729 


730 


532900 


389017000 


27-018512 


9-00411 


730 


731 


534361 


390617891 


27-037011 


9-00822 


731 


732 


535824 


392223168 


27-055498 


9-01232 


732 


733 


537289 


393832837 


27-073972 


9-01643 


733 


734 


538756 


395446904 


27-092434 


9-02052 


734 


735 


540225 


397065375 


27-110883 


9-02462 


735 


736 


541696 


398688256 


27-129319 


9-02871 


736 


737 


543169 


400315553 


27-147743 


9-03280 


737 


738 


544644 


401947272 


27-166155 


9-03688 


738 


739 1 


516121 


403583419 


27-184554 


9 04096 


739 


740 1 


547600 


405224000 


27-202941 


9-04504 


740 


741 


51^081 


406869021 


27-221315 


9-049U 


741 


742 


550564 


408518488 


27-239676 


9-05318 


742 


743 


552049 


410172407 


27-258026 


9-05724 


743 


744 


553536 


411830784 


27-276363 


9-06130 


744 


745 


555025 


413493625 


27-294688 


9-06536 


745 



1?6 



Squares, Ctbes, Sqcaee Roots, axd Ccbe Root?. 



. Number 

1 


Square. 
J 


Cube. 


' Sqaare Root 


CuheRoot, 


Numbef. 


I 746 


556516 


4151609S6 


27-313000 


j 9-06942 


746 


j 747 


55S009 


U6So2-\ld 


27-331301 


! 9-07347 


I ''^ 


] 718 


559504 


4185089y2 


^ 27-349588 


9-07751 


74S 


749 


561001 


420189749 


1 27-367864 


908156 


74f 


750 


562500 


421875000 


;i 27-386127 


9-08560 


750 


751 


564001 


* 423564751 


27-404379 


9-08963 


1 "^51 


752 


565504 


I 425259008 


27-422618 


9-09367 


1 "^5^ 


753 


567009 


1 426957777 


27-440845 


9O9770 


733 


754 


568516 


428661064 


27-459060 


910172 


754 


755 


570025 


430368875 


27-477263 


9-10574 


755 


756 


571536 


4320S1216 


27-495454 


910976 


756 


757 


573049 


! 433798093 


i 27-513633 


911378 


757 


758 


574564 


1 435519512 


! 27-531799 


911779 


758 


759 


576081 


1 437245479 


I 27-549954 


9-1-2180 


I 759 


760 


577600 


3 438976000 


27-568097 


9-12580 


j 760 


761 


579121 


440711081 


27-586228 


9-12980 


761 


762 


530644 


442450728 


27-604347 


9-13380 


762 


76S 


582169 


444194947 


27-622454 


9-13779 


1 763 


764 


583696 


445943744 


27-640549 


9-14178 


1764 


765 


585225 


447697125 


27-658633 


9-14577 


1 ^^^5 


766 


586756 


449455096 


27-676705 


9-14975 


i 76$ 


767 


588289 


451217663 


27-694764 


9-15373 


i '^^ 


768 


589824 


452984832 


27-712812 


9-15771 


\ 7ea 


769 


591361 


454756609 


27-730849 


916168 


rG9 


770 


592900 


456533000 


27-748873 


9-16565 


] 770 


771 


594441 


458314011 


27-766886 


9-16962 


771 


772 


595984 


460099648 


27'7S48S8 


9-1735B 


m 


773 


597529 


461889917 


27'js.j2S77 


917754 


7n 


774 


599076 


463684824 


27-820855 


9-l«150 


774 


775 


600625 


465484375 


27-838821 


9-18545 


775 


776 


602176 


467288576 


27-856776 


9-18940 


776 


777 


603729 


469097433 


27-874719 


9-19334 


777 


778 


605284 


470910952 


27-892651 


9-19728 


778 


7T9 


606841 


472729139 


27-910571 


9-20122 


779 


780 


608400 


474552000 


27-928480 


9-20516 


78a 


781 


609961 


476379541 


27-946377 


9-20909 


78i 


782 


611524 


478211768 


27-964262 


9-21302 


78« 


783 


613089 


4^«»48687 


27-982137 


9-21695 


78ir 


784 


614656 


481890304 


28-0 


9-22087 


784 


785 


616225 


483736625 


28-017851 


9-22479 


785 


786 


617796 


485587656 


28-035691 


9-22870 


78© 


787 


619369 


487443403 


28-053520 


9-23261 


78t 


788 


620944 


489303872 


28-071337 


9-23652 


78S 


789 


622521 


491169069 


28-089143 


9-24043 


789 


790 


624100 


493039000 


28-106938 


9-24433 


790 


791 


625681 


494913671 


28124722 


9-24823 


791 


792 


627264 


496793088 


28142494 


9-52213 


792 













Squares, Cubes, Square Roots, and Cube Roots. 



19Y 



Number. 


Square. 


Cube. 


Square Root. Cube Root. 


Number. 


793 


628849 


498677257 


28-160255 


9-25602 


793 


794 


630436 


500566184 


28-178005 


9-25991 


794 


795 


632025 


502459875 


28-195744 


9-26379 


795 


796 


633616 


504358336 


28-213472 


9-26767 


796 


797 


635209 


506261573 


28-231188 


9-27155 


797 


798 


636804 


508169592 


28-248893 


9-27543 


798 


799 


638401 


610082399 


28-266588 


9-27930 


799 


800 


640000 


512000000 


28-284271 


9-28317 


800 


801 


641601 


513922401 


28-301943 


9-28704 


801 


802 


643204 


515849608 


28-319604 


9-29090 


802 


803 


644809 


517781627 


28 -.337 254 


9-29476 


803 


804 


646416 


519718464 


28-354893 


9-29862 


804 


805 


648025 


521660125 


28-372521 


9-30247 


805 


806 


649636 


523606616 


28-390139 


9-30632 


806 


807 


651219 


525557943 


28-407745 


9-31017 


807 


808 


652864 


527514112 


28-425340 


931401 


808 


809 


654481 


529475129 


28-442925 


9-31785 


809 


810 


656100 


531441000 


28-4 60498 


9 32169 


810 


811 


657721 


533411731 


28-478061 


9-32553 


811 


812 


659344 


535387328 


28-495613 


9-32936 


812 


813 


660969 


537367797 


28-513154 


9-33319 


813 


814 


662596 


539353144 


28-530685 


9-33701 


814 


815 


664225 


541343375 


28-548204 


9 34083 


815 


816 


665856 


543338496 


28-565713 


9-34465 


816 


817 


667489 


545338513 


28-583211 


9-34847 


817 


818 


669124 


547343432 


28600699 


9-35228 


818 


819 


670761 


549353259 


28-618176 


9-35609 


819 


820 


672400 


551368000 


28-635642 


9-35990 


820 


821 


674041 


553387661 


28-653097 


9-36370 


821 


822 


675684 


555412258 


28-670542 


9-36750 


822 


823 


677329 


557441767 


28-687976 


9-37130 


823 


824 


678976 


559476224 


28-705400 


9-37509 


824 


825 


680625 


561515625 


28-722813 


9-37888 


825 


826 


682276 


563559976 


28-740215 


9-38267 


826 


827 


683929 


565609283 


28-757607 


9-38646 


827 


828 


685584 


567663552 


28-774989 


9-39024 


828 


829 


687241 


569722789 


28-792360 


9-39402 


829 


830 


688900 


571787000 


28-809720 


9-39779 


830 


831 


690561 


573856191 


28-827070 


9-40156 


831 


832 


692224 


575930368 


28-844410 


9-40533 


832 


833 


693889 


578009537 


28861739 


9-40910 


833 


834 


695556 


580093704 


28-879058 


9-41286 


834 


835 


697225 


582182875 


28-896366 


9-41662 


835 


836 


698896 


584277056 


28-913664 


9-42038 


836 


837 


700569 


586376253 


28-930952 


9-42414 


837 


838 


702244 


588480472 


28-948229 


9-42789 


838 


839 


703921 


590589719 


28-965496 


9-43164 


839 



17^ 



198 


Squares, 


Cubes, Squar 


E Roots, and Cube Roots. 




Number. 


Square. 


Cube. 


Square Root. ( 


:ube Root. 


Number. 


840 


705600 


592704000 


28-982753 


9-43538 


840 


841 


707281 


594823321 


29-0 


9-43913 


841 


842 


708964 


596947688 


29-017236 


9-44287 


842 


843 


710649 


599077107 


29-034462 


9-44660 


843 


844 


712336 


601211584 


29-051678 


9-45034 


844 


845 


714025 


603351125 


29-068883 


9-45407 


845 


846 


715716 


605495736 


29-086079 


9-45779 


846 


847 


717409 


607645423 


29103264 


9-46152 


847 


848 


719104 


609800192 


29-120439 


9-46524 


848 


849 


720801 


611960049 


29-137604 


9-46896 


849 


850 


722500 


614125000 


29-154759 


9-47268 


850 


851 


724201 


616295051 


29-171904 


9-47639 


851 


852 


725904 


618470208 


29-189039 


9-48010 


852 


853 


727609 


620650477 


29-206163 


9-48381 


853 


854 


729316 


622835864 


29-223278 


9-48751 


854 


855 


731025 


625026375 


29-240380 


9-49122 


855 


856 


732736 


627222016 


29-257477 


9-49491 


856 


857 


734449 


629422793 


29-274562 


9-49861 


857 


858 


736164 


631628712 


29-291637 


9-50230 


858 


859 


737881 


633839779 


29-308701 


9-50599 


859 


860 


739600 


636056000 


29-325756 


9-50968 


860 


861 


741321 


638277381 


29-342801 


9-51336 


861 


862 


743044 


640503928 


29-359836 


9-51705 


862 


863 


744769 


642735647 


29-376861 


9-52073 


863 


864 


746496 


644972544 


29-393876 


9-52440 


864 


865 


748225 


647214625 


29-410882 


9-52807 


865 


866 


749956 


649461896 


29-427877 


9-53174 


866 


867 


751689 


651714363 


29-444863 


9-53541 


867 


868 


753424 


653972032 


29-461839 


9-53908 


868 


869 


755161 


656234909 


29-478805 


9 '54274 


869 


870 


756900 


658503000 


29-495762 


9-54640 


870 


871 


758641 


660776311 


29-512709 


9-55005 


871 


872 


760384 


663054848 


29-529646 


9-55371 


872 


873 


762129 


665338617 


29-546573 


9 55736 


873 


874 


763876 


667627624 


29-563491 


9-56101 


874 


875 


765625 


669921875 


29-580398 


9-56465 


875 


876 


767376 


672221376 


29-597297 


9-56829 


876 


877 


769129 


674526133 


29-614185 


9-57193 


877 


878 


770884 


676836152 


29-631064 


9-57557 


878 


879 


772641 


679151439 


29-647934 


9-57920 


879 


880 


774400 


681472000 


29-664793 


9-58283 


880 


881 


776161 


683797841 


29-681644 


9-58646 


881 


882 


777924 


686128968 


29-698484 


9-59009 


882 


883 


779689 


688465387 


29-715315 


9-59371 


883 


884 


781456 


690807104 


29-732137 


9-59733 


884 


885 


783225 


693154125 


29-748949 


9-60095 


885 


886 


784996 


695506456 


29-765752 


9-60456 


886 



Squares, Cubes, Square Roots, and Cube Roots. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Number. 


887 


786769 


697864103 


29-782545 


9-60818 


887 


888 


788544 


700227072 


29-799328 


9-61179 


888 


889 


790321 


702595869 


29-816103 


9-61539 


889 


890 


792100 


704969000 


29-832867 


9-61900 


890 


891 


793881 


707347971 


29-849623 


9-62260 


891 


892 


795664 


709732288 


29-866369 


9-62620 


892 


893 


797449 


712121957 


29-883105 


9-62979 


893 


894 


799236 


714516984 


29-899832 


9-63339 


894 


895 


801025 


716917375 


29-916550; 


9-63698 


895 


896 


802816 


719323136 


29-933259 


9-64056 


896 


897 


804609 


721734273 


29-949958 


9-64415 


897 


898 


806404 


724150792 


29-966648 


9-64773 


898 


899 


808201 


726572699 


29-983328 


9-65131 


899 


900 


810000 


729000000 


30-0 


9-65489 


900 


901 


811801 


731432701 


30-016662 


9-65846 


901 


902 


813604 


733870808 


30-033314 


9-66204 


902 


903 


815409 


736314327 


30-049958 


9-66560 


903 


904 


817216 


738763264 


30-066592 


9-66917 


904 


905 


819025 


741217625 


30-083217 


9-67274 


905 


906 


820836 


743677416 


30-099833 


9-67630 


906 


907 


822649 


746142643 


30-116440 


9-67986 


907 


908 


824484 


748613"312 


30-133038 


9-68341 


908 


909 


826281 


751089429 


30-149626 


9-68697 


909 


910 


828100 


753571000 


30-166206 


9-69052 


910 1 


911 


829921 


756058031 


30-182776 


9-69406 


911 


912 


831744 


758550528 


30-199337 


9-69761 


912 


913 


833569 


761048497 


30-215889 


9-70115 


913 


914 


835396 


763551944 


30-232432 


9-70469 


914 


915 


837225 


766060875 


30-248966 


9-70823 


915 


916 


839056 


768575296 


30-265491 


9-71177 


916 


917 


840889 


771095213 


30-282007 


9 71530 


917 


918 


842724 


773620632 


30-298514 


9-71883 


918 


919 


844561 


776151559 


30-315012 


9-72236 


919 


920 


846400 


778688000 


30-331501 


9-72588 


920 


921 


848241 


781229961 


30-347981 


9-72941 


921 


922 


850084 


783777448 


30-364452 


9-73293 


922 


923 


851929 


786330467 


30-380915 


9-73644 


923 


924 


853776 


788889024 


30-397368 


9-73996 


924 


925 


855625 


791453125 


30-413812 


9-74347 


925 


926 


857476 


794022776 


30-430248 


9-74698 


926 


927 


859329 


796597983 


30-446674 


9-75049 


927 


928 


861184 


799178752 


30-463092 


9-75399 


928 


929 


863041 


801765089 


30-479501 


9-75750 


929 


930 


864900 


804357000 


30-495901 


9-76100 


930 


931 


866761 


806954491 


30-512292 


9-76449 


931 


932 


868624 


809557568 


30-528675 


9-76799 


932 


933 


870489 


812166237 


30-545048 


9-77148 


988 



200 



Squares, Cubes, Square Roots, and Cube Roots. 



Number 

1 


Square. 


Cube. 


Square Root. 


Cube Root. 


Number. 


934 


872356 


814780504 


80-561413 


9-77497 


934 


985 


874225 


817400375 


30-577769 


9-77846 


935 


936 


876096 


820025856 


30-594117 


9-78194 


986 


937 


877969 


822656953 


30-610455 


9-78542 


937 


938 


879844 


825293672 


30-626785 


9-78890 


938 


939 


881721 


827936019 


30-643106 


9 79288 


989 


940 


883600 


830584000 


30-659419 


9-79586 


940 


941 


885481 


833237621 


30-675723 


9-79938 


941 


942 


887364 


835896888 


30-692018 


9-80280 


942 


943 


889249 


888561807 


80-708305 


9-80627 


943 


944 


891136 


841232384 


30-724583 


9-80978 


944 


945 


893025 


843908625 


30-740852 


9-81819 


945 


946 


894916 


846590536 


30-757113 


9-81665 


946 


947 


896809 


849278123 


30-773365 


9-82011 


947 


948 


898704 


851971392 


30-789608 


9-82357 


948 


949 


900601 


854670349 


30-805843 


9-82702 


949 


950 


902500 


857375000 


30-822070 


9-83047 


950 


951 


904401 


860085351 


30-888287 


9-83392 


951 


952 


906304 


862801408 


30-854497 


9-83786 


952 


953 


908209 


865523177 


30-870698 


9-84081 


953 


954 


910116 


868250664 


30-886890 


9-84425 


954 


955 


912025 


870983875 


30-903074 


984769 


955 


956 


913936 


873722816 


80-919249 


9-85112 


956 


957 


915849 


876467493 


30-985416 


9-85456 


957 


958 


917764 


879217912 


30-951575 


9 85799 


958 


959 


916681 


881974079 


30-967725 


9-86142 


959 


960 


921300 


884736000 


30-983866 


9-86484 


960 


961 


923521 


887503681 


31-0 


9-86827 


961 


962 


925444 


8902771.28 


81016124 


9-87169 


962 


963 


927369 


893056347 


31-082241 


9-87511 


963 


964 


929296 


895841844 


31-048349 


9 87853 


964 


965 1 


981225 


898682125 


31-064449 


9-88194 


965 


966 I 


938156 


901428696 


81-080540 


9-88535 


966 


967 


935089 


904281063 


81-996623 


9-88876 


967 


968 1 


937024 


907039282 


81-112698 


9-89217 


968 


969 i 


938961 


909858209 


81-128764 


9-89558 


969 


970 


940900 


912673000 


. 81-144828 


9-89898 


970 


971 


942841 


915498611 


31-160872 


9-90238 


971 


972 


944784 


918330048 


31-176914 


9-90578 


972 


973 i 


946729 


921167317 


31-192947 


9-90917 


973 


974 i 


948676 


924010424 


31-208973 


9 91257 


974 


975 ; 


950625 


926859875 


31224990 


9-91596 


975 


976 ' 


952576 


929714176 


31-240998 


9-91935 


976 


977 1 


954529 


982574838 


81-256999 


9-92278 


977 


978 


956484 


935441352 


31-272991 


9-92612 


978 


979 


958441 


938813789 


81-288975 


9-92950 


979 


980 


960700 


951192000 


31-304951 


9-93288 


980 



Squares, Cubes, Square Roots, and Cube Roots. 



201 



iN umber. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Number. 


981 


962361 


944076141 


31-320919 


9-93626 


981 


982 


964324 


946966168 


81-336879 


9-98963 


982 


988 


966289 


949862087 


31-352830 


9 94300 


983 


98i 


968256 


952763904 


81-368774 


9-94687 


984 


985 


970225 


955671625 


31-384709 


9-94974 


985 


986 


972196 


958585256 


81-400636 


9-95311 


986 


987 


974169 


961504803 


31-416556 


9-95647 


987 


988 


976144 


964430272 


31-432467 


9-95983 


988 


989 


978121 


967361669 


81-448370 


9-96819 


989 


990 


980100 


970299000 


31-464265 


9-96655 


990 


991 


982081 


973242271 


31-480152 


9-96990 


991 


992 


984064 


976191488 


81-496031 


9-97326 


992 


993 


986049 


979146657 


81-511902 


9-97661 


993 


994 


988036 


982107784 


31-527765 


9-97995 


994 


995 


990025 


985074875 


81-543620 


9-98330 


995 


996 


992016 


988047936 


81-559467 


9-98664 


996 


997 


994009 


991026973 


31-575306 


9-98999 


997 


998 


996004 


994011992 


81-591138 


9-99332 


998 


999 


998001 


997002999 


31-606961 


9-99666 : 


999 


1000 


1000000 


1000000000 


81-623776 


10- 


1000 















Silver, to Purify and Reduce. — Silver, as used in the arts and 
coinage, is alloyed with a portion of copper. To purify it, dissolve 
the metal m nitric acid slightly diluted, and add common salt, 
which throws down the whole of the silver in the form of chloride. 
To reduce it into a metallic state several methods are used: 1. 
The chloride must be repeatedly washed with distilled water, and 
placed in a zinc cup; a little diluted sulphuric acid being added, 
the chloride is soon reduced. The silver when thoroughly washed 
is quite pure. In the absence of a zinc cup, a porcelain cup con- 
taining a zinc plate may be used. The process is expedited by 
warming the cup. 

2. Digest the washed chloride with pure copper and ammonia. 
The quantity of ammonia need not be sufficient to dissolve the 
chloride. Leave the mixture for a day, then w^ash the silver 
thoroughly. 

3. Boil the washed and moist chloride in solution of pure potash, 
adding a little sugar : when washed it is quite pure. 

Welding Composition. — Mix borax with y'^th of sal ammoniac, 
fuse the mixture, and pour it on an iron plate. When cold, pul- 
verise it, and mix it with an equal weight of quick lime, sprinkle it 
on iron, which is heated to redness, and replace it in the tire. It 
may be welded below the usual lieat. 



202 Blacking Recipes. 



BLACKIXG EECIPES. 

Liquid Blacking, for Boots and Shoes. — 1. Ivory black, 3 oz. ; 
molasses, 2 oz. ; sweet oil, \ oz. Mix to form a paste. Add gradu- 
ally ^ oz. of oil of vitriol, and then half a pint of vinegar, and If 
piat of water, or sour beer. fcome prefer mixing the oil of vitriol 
with the sweet oil. 

2. Ivory black, 2 lbs.; molasses, 2 lbs. ; sweet oil, ^ lb. Mix, and 
add f lb. oil of vitriol, and enough beer or vinegar to make up a 
gallon. 

3. Ivory black, 3 lbs. ; molasses, 4 lbs. ; vinegar, 1 piat; oil of 
vitriol, S oz. ; water, 1 gallon. 

4. Ivory black, 2 lbs. ; neat's-foot oil, 4 oz. Mix, and add 3 quarts 
of sour beer or vinegar, and a spoonful of any kind of spirits ; stir 
till smooth, and add 2 oz. of oil of vitriol, and sprinkle on it ^ 
drachm of powdered resin. Then boil together 3 pints of sour ale 
with a little logwood, and ^ oz. of Prussian blue, 3 oz. of honey, 
and 8 oz. of molasses. Mix, but do not bottle it for two or three 
days. 

0. Ivory black, 8 oz. ; brown sugar, or molasses, 8 oz. ; sweet oil, 
1 oz. ; oil of vitriol, ^ oz. ; vinegar, two quarts. Mix the oil with 
the molasses, then add the oil of vitriol and vinegar, and lastly the 
ivory black. 

Blacking for Dress Boots — 1. Gum, 8 oz. ; molasses, 2 oz. ; ink, 1 
pint ; vinegar, 2 oz. ; spirit of wine, 2 oz. Dissolve the gum and 
molasses in the ink and vinegar, strain, and add the spirit. 
I 2. To the above add 1 oz. of sweet oil, and ^ oz. of lampblack. 
i [These are applied with a sponge, and allowed to dry out of the 
i dust. They will not bear the wet] : 

i 3. Beat together the whites of 2 eggs, a table-spoonful of spirit ; 
i of wine, a lump of sugar, and a little finely powdered ivor^- black ; 
i to thicken. I 

j Blacking, icithout Polishing. — Molasses, 4 oz. ; lampblack, ^ oz. ; ! 
3'east, a table-spoonful ; 2 eggs ; a tea-spoonful of olive oil ; a tea- j 
spoonful of turpentine. Mix well. To be applied with a sponge, 
without brushing. 

India Rubber Blacking.— lyovy blnck, 60 lbs. ; molasses, 45 lbs. ; 
vinegar (No. 24), 20 gallons ; powdered gum, 1 lb. ; India rubber 
oil, 9 lbs. (The latter is made bv dissolvinfr, bv heat, 18 oz. of 
India rubber in 9 lbs. of rape oil,) ^ Grind the whole smooth in a 
paint mill. Then add, by small quantities at a time, 12 lbs. of oil 
I of vitriol, stirring it stronsrly for half an hour a day for a fortnight. 
I Paste Blacking. — \. Oil of vitriol, 2 parts; sweet oil, 1 part; 
'. molasses, 3 parts ; ivory black, 4 parts. Mix. 

1 2. This may be made with the ingredients of liquid blacking, 
; using sufficient vinegar, in which a little gum has been dissolved, 
■ to form a paste. Make it into cakes, and dry it. 



Blacking Recipes. 



203 



3. (Bailey's Blacking Balls.) Bruised gum tragacantli, 1 oz. ; 
water, 4 oz. Mix, and add 2 oz. of neat's-foot oil, 2 oz. of fine iyovj 
black, 2 oz. of Prussian blue. Mix, and evaporate to a proper con- 
sistence. 

Blacking for Hartiess. — 1. Isinglass or gelatine, ^ oz. ; powdered 
indigo, ^ oz ; soft soap, 4 oz. ; logwood, 4 oz. ; glue, 5 oz. Boil 
together in 2 pints of vinegar, till the glue is dissolved; then strain 
through a cloth, and bottle for use. 

2. Melt 8 oz. of beeswax in an earthen pipkin, and stir into it 2 
oz. of ivory black, 1 oz. of Prussian blue ground in oil, 1 oz. of oil 
of turpentine, and ^ oz. of copal varnish. Make it into balls. To 
be applied with a brush, and polished with an old handkerchief 

3. Molasses | lb. ; lampblack, 1 oz. ; yeast, 1 spoonful ; of sugar 
candy, olive oil, gum tragacanth, and isinglass, 1 oz. each ; a cow's 
gall. Mix all together with 2 pints of stale beer, and let it stand 
before the fire for an hour. 

Heel Balls. — 1. Melt together 4 oz. of mutton suet, 1 oz. of bees- 
wax, 1 oz. of sweet oil, -^ oz. oil of turpentine, and stir in 1 oz. of 
powdered gum arable, and ^ oz. of fine lampblack. 

2. Beeswax, 8 oz. ; tallow, 1 oz. ; powdered gum, 1 oz. ; lamp- 
black, q. s. 

Heel balls are used not merely by the shoemaker, but to copy 
inscriptions, raised patterns, <fec., by rubbing the ball on paper laid 
over the article to be copied. 



Black LEAD Pencils. — The easiest way of producing, not only 
blacklead, but all sorts of pencils, is by the following process, which 
at once combines simplicity, cheapness, and the finest qualit}'. 

Take white or pipe-clay : put it into a tub of clean water, to soak 
for twelve hours, then agitate the Avhole, until it resembles milk, 
let it rest two or three minutes, and pour off the supernatant 
milky liquor into a second vessel, allow it to settle, pour off the 
clear, and dry the residue on a filter. Then add blacklead, any 
quantity. Powder it, and calcine it at a white heat in a loosely 
covered crucible, cool, and carefully pulverize, then add prepared 
clay, prepared plumbago, equal parts. Water to mix. Make them 
into a paste, and put it into oiled moulds of the size required, dry 
very gradually, and apply sufficient heat to give the required 
degree of hardness; lastly, the pieces should be taken carefull}' 
from the moulds, and placed in the gioovesof the cedar. The more 
clay and heat employed the harder the crayon ; less clay and heat 
of course produces a contrary effect. The shade of black may also 
be varied in tlie same way. Each mould must be made of four 
pieces of wood, nicely fitted together. 

Black for Miniature Painters. — Take camphor, and set it on llu^ 
fire, and collect the soot by means of a saucer or paper funnel 
mverted over it. 



204 



Strain and Stress of Materials. 



STRAIX AXD STEESS OF 



MATEPJALS. 

Fig. 1. 






D 



Let A B he a beam of timber, firmly fix- 
ed in a wall at A, and a ^veight, TF, mea- 
,siired in pounds avoirdupois, acting at the 
extremity B, at right angles to A B. 

If A 5 be one foot, and the av eight IF be 
one pound, then the strain produced at^ is 
called a unit of strain. 

If the beam ^ ^ be (/) feet long, and the 
weight be ( W) pounds, then the imits of 

strain produced at A, by the weight acting ^^*^W^ 

at B, will be I W. And the units of strain ^^^^ 

which the weight W produces at any other part of the beam i>, are 
measured by IF. B D. 

Let J. ^ = 10 feet, and the weight IF be equal to 112 lbs., and 
BD = 1 feet. 

Tlie units of strain at ^ = 112 x 10 = 1120. 

The u?nts of strain at D = 112 x 7 = '784. 

The greatest strain on the beam is at^, at which place the beam 
would break if it was equally strong throughout 

If the weight TFbe uniformly distributed ^^' * 

over the whole length of the beam A B, as 
in fig. 2, the fuiits of strain at A will be 
only one-half as great as that produced by ^%^ 
the weight TF acting as in fig. 1. " '" ' 

Tlie units of strain at A^ which are pro- 
duced by the beam itself, are equal to the 
weight of the beam multiplied by half its length. 

The beam A B, fig. 3, is equally 
sti*ong between the points A and B, 
when the undei'^ide of it is a common 
parabola. 

Hence, from a square beam, one-third 
part of it may be cut off without di- 
minishing its strength. 



If the weight IF be uniformly distri- 
buted over "the whole length of the i 
beam A B, as in fig. 4; then the beam 
is equally strong when the underside of 
it is a straight line. In this case, one 
half the beam may be cut away without 






diminishing its strength. 



©TRAIN AND STRESS OF MATERIALS. 



205 



Fig. 5 
G 



^A 



Let the Aveight W (fig. 5) be sustained by a beam A B, which 
rests on two props at C and D. 

The pressure on the prop at C is equal to W . BE: AB. 

The pressure on the prop at D is equal to W. A E : A B. 

The units of strain at E are equal to W . A^ E . B E : A B. 

The units of strain at G are equal to W . A E. B G : AB. 

The units of strain at i^are equal to W . B E . A E : A B. 

The greatest strain, which is produced by the weight W, is at E. 

The units of strain at the middle of the beam, produced by the 

W. AE 
weight TFactmg at E, are equal to * 

Let ^ B = 18 feet, and a weight of 112 lbs. be placed at E, 

which is 8 feet from A. 

Apply these numbers to the above formulae and their results. 

10 X 112 

The pressure on the prop at C is equal to = 62.5 lbs. 

1 8 

8 X 112 
The pressure on the prop at I) is equal to — = 49.8 lbs. 

lo 

^^ . ^ , 10 X 8 X 112 ^ ^. 

The U7iits of strain at E are equal to -— = 497.77. 

8 X 112 
The units of strain on the middle are equal to — — = 448. 

When the weight W is laid on the middle of the beam A B, the 

. ^ . 1^ ..n. 1 W.AB 
units of strain on the middle are equal to * 

If the weight W be uniformly distributed along the beam A By 

W. A B 

the units of strain on the middle of it will be equal to — ' ; 

8 
which is only one half the strain that is produced by the weight 
having been laid on the middle. 



■^^ 



m^ Fig. 6. 



18 



206 



Strain and Stress of Materials. 



When the beam A B (fig. 6), supports a weight ir, at E, it is 
equally strong between the points A and B, if the upper sides, 
A E, BE, be two parabolas whose vertex is A and B respectively. 

Fig. 7. 



^v 



\ 



Let the weight TT^have a bearing^ J' (fig. T), equal on both sides j 
of the centre G, and also let the weight be equaUy distributed on j 
the bearing E F. \ 

TT" 4 B TT^ E E ^ 

The units of strain at G are equal to — — ^ ! 

!N"ow, if the weight TT were a sphere, and were laid on the mid- 
dle of the beam at G, the units of strain at G would be equal to 
W. A B 



::<.^yb^: 



Kthe same weight be formed into a cube, whose side is E E, the 

units of straiji at the centre 6^ will be less than in the ease of the 

^ W. EF 
sphere bv . 

^ Fig. 8. J. 

Let ^ ^ be any beam suspended vertically from the -^7.^ 

point A (fig. S) : and let the sectional area be constant 
from A to B, where a weight TT lbs, is acting to 
extend the beam. 

Put a = area of tlie section of the beam in square 
inches. 
I = length of the beam in feet before the 

weight is applied to elongate it. 

€ = the elongation produced by the weight W. 

E= wei^rht which would be necessary to make 

tf equal to /. The quantity ^is called the modulus 

of elasticity of the material of which the beam is composed. 

In the case of the beam being compressed by the weight W act- 
ing in the opp»osite direction, 

Put c = compression produced by the weight TT. 

C = force which is necessary to make c equal to half of (I). 
The quantity C is called the modulus of elasticity of 
the material, when it is subject to compression. 



LIB 



nr 



E = 



WI 



and = 



W! 



Units of work done to elonsrate the b 



^am € leet = — — . 



Strain and Stress of Materials. 



207 



Wc 
Units of work done to compress the beam c feet = . 



Mean results of experiyne^its on four different hinds of Cast-iron bars, 
iO feet long and 1 square inch in section. 



"Weight laid on bar 

per square inch 

- W. 



lbs. 

1054 

1581 

2108 

3161 

4215 

5269 

6323 

7376 

8430 

9484 

10538 

11591 

12645 

13700 

14793 



Extension of bar in 
inches := 12 e. 



inches. 

.009 

.0137 

.0186 

.0287 

.0391 

.0500 

.0613 

.0734 

.0859 

.0995 

.1136 

.1283 

.1448 

.1668 

.1859 



Set of bar in inches. 


]2 W 

The value of . 

e 




117085 


,00022 


115131 


.00055 


113308 


.00107 


110150 


.00175 


107802 


.00265 


105377 


.00372 


103142 


.00517 


100496 


.00664 


98,139 


.00844 


95316 


.01062 


92762 


.01306 


90347 


.01609 


87329 


.02097 


82133 


.02410 


7fi576 



Hence, the breaking weight per square inch of section is 14793 
lbs. = 6.6 tons nearly; and the ultimate extension is .1859 inches, 
or gl^ of the whole length, 10 feet. 

If we deduct the set -0209 from -1859, we shall have 165 inches 
for the elongation produced by the weight 14793 lbs. 

.... 14793 X 10 X 12 

.*. E= modulus of elasticity — "ttt^ = 10758545. 

.loo 

.*. Breaking weight = 6.6 tons x area of sectioi\ in square inches. 

If the weight 5269 be taken, the modulus of elasticity will be 

considerably increased. Deduct .00175 the set from .05, leaving 

.04825 inches for the elongation due to the Aveight 5269 lbs. 

59^9 X 10 X 12 

.\E= modulus of elasticity rr ---^ = 13104249. 

■^ .04825 

This difference in tlie modulus of elasticity arises from the cir- 
cumstance of the law of elasticity not being proportional to the 
weight. 



208 



Stsaix and Stress of Materials. 



TABLE 

Of the Tensile Strength of Wrought Iron. 
The Bar was 10 feet long and 1 square inch section. 



Weight laid on the 
Bar W. 


Extension of the Bar 
or value of 12 e. 


Set of Bar. 


The value of 
12 W 

€ 


lbs. 


inches. 


inches. 




1262 


•00520 




242665 


S'FSS 


•01690 


•0005 


223998 


6309 


• 0-2772 


•0005 


227608 


8833 


•03790 


•0005 


233061 


11356 


•04854 


•0005 


233966 


13880 


•05950 


•0007 


233285 


16404 


•06980 


•0007 


235016 


18928 


•08170 


•00130 


231675 


21452 


•09310 


•00270 


230415 


23975 


•10570 


•00410 


226824 


23499 


•12040 


•00680 


220092 


29023 


•14500 


•0120 


200157 


30284 
31546 


•19910 
•28660 
•24200 


•0120 , 

•1082 \ 

•1083 


after bearing the 
weight 17 hours. 
130357 


ditto 


• 24490 


•nil 


after five minutes. 


35332 


2-04 


1-874 


17320 



The bar broke with a weight of 24 tons per square inch of 
section. Hence the tensile force of wrought iron is nearly four 
ticQes as great as the tensile force of cast iron. 

TABLE 

Of the Compresswe Strength of Wrought Iron, 
The Bar was 10 feet long and I square inch section. 



Weight laid on the 
Bar, or (jy). 



Decrement of length, 
or the val ue of 12 c. 



Weieht laid on the 
bar, or(jr). 



Decrement of length, or 
the value of 12c. 



lbs. 

5098 
9578 
14058 
16298 
18538 
20778 



inches. 

•028 
•052 
•073 
•085 
•096 
•107 



lbs. 

23018 
25258 
27498 
29738 
31978 
34218 



•119 
•130 
•142 
•154 
•174 
•214 



Strain and Stress of Materials. 



209 



The crushing force of wrought iron is 12 tons per square inch. 
It is a curious fact, that cast iron is decreased in length nearly 
double what wrought iron is, by the same weight ; but the 
wrought iron bar will sink to any degree with little more than 12 
tons per square inch, whilst cast iron will bear 43*56 tons to 
produce the same effect. 

A ^vrought bar will bear a compression of ^Jg- of its length, 
without its utility being destroyed. 

Conipressioii of Cast Iron. 
Mean results of experiments on four different kinds of Cast Iron, lo feet lon^, and 1 
square inch in section. ,, 



Weight laid on the 


Decrement of length, 




The value of 
12 W 


bar ( [VI. 


or the value ot \z c. 


Set of bar in niches. 


c 


lbs. 


mches. 


mches. 




2065 


•01875 


•00047 


110119 


4129 


•03878 


•00226 


106485 


6194 


•05978 


•00400 


103617 


8259 


•07879 


•00645 


104822 


10824 


•09944 


•00847 


103819 


12388 


•12030 


•010875 


102980 


14453 


•14163 


•01405 


102049 


16518 


•16338 


•01712 


101101 


18583 


•18505 


•02051 


100420 


20464 


•20624 


•02484 


100114 


24777 


•24961 


•03220 


99263 


28906 


•29699 


•04300 


97331 


33031 


•35341 


•06096 


93463 



The crushing or compressive force of cast iron per square inch is 
43*56 tons, which has been obtained from eleven kinds of cast iron. 
But the tensile force of cast iron is 6 "6 tons ; therefore the compres- 
sive force is equal to the square of the tensile force, or (6 '6)^. 



Transvers 




Strength of Beams. 

To find the neutral line, forces of 
extension, forces of compression, mo- 
ments of extension, and moments of 
compression of a beam subject to 
transverse flexure. 

Let the form of the section of the 
beam be that of the figure A B D E, 
where BC, HE, represent sections of 
the top and bottom ribs, EG that of 
the vertical one connecting them, and 
NO pass through the neutral line. 

Put a, a' = N I, NK, respectively. 

18* 



210 



Strain and Stress of Materials. 



c, c' = D ff, A C, respectively. 
b,b' =.DB,AB, do. 
/? = the thickness of the vertical rib. 
/,/'= tensile and compressive forces of the material, in a square 
inch of section, as exerted at a distance (a) on opposite sides of the 
neutral line. 

For the determination of the neutral line 

/ I 5a'_ (6_,3) (a-cf \=f'\f>' a'=- {b'-6) {a'-df | 

And a -^ a' =z D, where D is the whole depth of the beam. 
For moderate strains per square inch/=/' 

... ha'-^b—S) {a—cf=b' (JD—af—{b'—p){D—a—cy 

Moments of extension = — -j ba^ — {b — /?) (a — cf V 

Moments of compression = ^\ b' a'^ — (6' — P) (a! — c'Y {• 
8 a ( J 

If W be the weight laid on the middle, and I equal length 
between supports, 

Wl f i \ 

■'■-r = Za \ ba' + b'ci^-{h-ii){a-cf-{b'-li){a'-cy^ 

If the form of the section be this, 

Then b' = (3 
Therefore, for the neutral line 
ba' — {b—P) {a^cf = /? {D — af. 
Moment of extension 

Moment of compression = -=^— 

O €1 



E 



Y///y//A///////////\ 

i 



1 






o 



4 3a ( ) 

If the form of the section be this, 

Then 6 = /? 
Therefore, for the neutral line 

/? a^ = 5' (i) — a)'^ — (6' — /?) (i) — rt — c)'' 

Moment of extension = - 



Strain and Stress of Materials. 



211 



Moment of compression 

And — = -^ j /? a' + 6' a"— (6' — 0) {a' — c'f i 

If the form of the section be this, 

Then, b = l3 and b' = p 
Therefore, for the neutral line 
2 a = J) 
or the neutral line is in the middle of the section. 

12 



Moment of extension = - 



i 



K 



Moment of compression = 



12 



.Wl=: 



2/1^ 



Transverse Strength of Cast-iron Bars. 



Length of Bar between supports, with its 
dimensions. 



Breaking 
weight laid 
on middle. 



Ultimate 

deflexion in 

inches. 



Mean of 

experi- 
ments. 



4^ feet, with 1 inch square 
9 feet, with 2 inches square 
13^ feet, with 3 inches square 
6f feet, with 3 inches square 



lbs. 
440 

1338 

2861 

611Y 



1'119 
3-0035 
4- 667 
1-2916 



2 f 
From the three last experiments we find — ^- = 1490. 



W= 1490 X 



0D^ 
I 



For a cast-iron beam, where W is the breaking weight in lbs., 
is the breadth of the beam measured in inches, Z> the depth of the 
beam measured in inches, and I the leiigtli of beam between sup- 
ports measured in feet. 



212 



Stkain and Stress of Materials. 




The best dimensions of a beam, -whose 
section is given in the figure, are when the 
bottom flange contains six times as much 
area as the top flange. And the breaking 
weight of such beams may be found by the 
following admirable rule: 

Multipl}^ the sectional area of the bottom 

flange in square inches, by the depth of the 

beam in inches, and divide the product by the distance between 

the supports, measured in feet, then 2*14 times the quotient will 

gi\'e the breaking weight in tons. 

A cast-iron bar is not weakened by passing half the breaking 
weight over it 96,000 times, with a velocity of 81 feet per minute. 

Defection of Beams. 

Let the beam be supported at A and B, and weight IF applied 
at the middle C. 




,DC = 



WP 



E-=^ the modulus of elasticity 

of the material. 
/? rr breadth of beam in in. 
D = depth of beam in inches. 
I =z length of beam in inches. 



Let the beam be supported at A, 
and a weight "FT applied at the other 
extremity. 

Eule for finding the ultimate de- 
flexion of a cast-iron beam : 



Ultimate deflexion D C in inches : 



Ultimate deflexion jS C in inches = 




for first figure, 
lor second figure. 



where I is measured in feet and D in inches. 

These values for the ultimate deflexion are independent of the 
breadth of the beam. 

Find the ultimate deflexion of a cast-iron bar, the distance 
between the supports being 24 feet, and depth 4^ inches. 



Strain and Stress of Materials. 



213 



Ultimate deflexion = zrx ~ 

40 D^ 



= 9G inches. 



40 X 4i 

If the weight Ti^be uniformly distributed along the beam, the 
deflexion will be in all cases f of the deflexion which is produced 
by the weight acting on the middle, or in the case of having only 
one support, acting at the extremity. 

Transverse Flexure of a Wrought-Iron Bar by Pressure acting 
Horizontally. 

Length of bar 14 feet 7^ inches, depth of bar in direction of 
pressure 1*515 inches, breadth 5523 inches, distance between 
supports 13 feet 6 inches. The experiment was continued to the 
limit of perfect elasticity, or to that point at which the elasticity 
was sensibly injured. 



Weight applied, actinp 


Deflexions after Sets 


after five 


Ratio of weights to 


horizontally. 


five minutes. rm 


inutes. 


deflexions. 


lbs. 


inches. i 


nches. 




28 


•051 


•0 


549 \ 


56 


•112 





500 




112 


•232 


•0 


483 




168 


•344 


001 


488 




224 


•458 


002 


489 




280 


•571 


003 


490 




336 


•684 


003 


491 






392 


•800 


004 


490 


, 




448 


•916 


006 


489 


00 

OS 




504 


1-005 


007 


501 






560 


1-124 


008 


498 




616 


1-222 


010 


504 


1^ 


672 


1-332 


Oil 


504 




728 


1-434 


017 


508 




784 


1-547 


019 


507 




840 


1-693 


019 


496 




896 


r823 


019 


492 




952 


1^933 


020 


493 




1008 


2^044 


021 


493 




1064 


2-165 


022 


491/ 



214 



Strain and Stress of Materials. 



To find the weight which a wrought-iron beam is capable of 
bearing without injuring its elasticity. 

tons, nearly. 



W=. ; • lbs. : 



I 



%l 



B and D are measured in inches, and I in feet, being the distance 
between the supports. 

What is the weight that can be laid on a wrought-iron bar, 20 
feet long, 3 inches broad, and 6 inches deep, without injuring its 
elasticity? ' ' '^' '''*' 

„^ 3 X 36 lOS 
.•.ir=-^^= -- = 2-7 tons. 

The deflexion of a wrought-iron beam, supported at each end, 
and loaded in the middle, when the elastic limit is obtained. 

Deflexion in inches =: -0167 x -jr. 

The length, I, is measured in feet, and i>, the depth, in inche& 
Taking the bar given in the last example, 

400 



Deflexion = -0167 



= 1-11 inches. 



v///////^///// ■/// /. 



ia, 



T> 



W////A/7/'/M 



Hollow Rectangular Beams. 

Let A B C B be the section of a hoUow rec- 
tangular beam. 

Let A I) = D, and ad = d 
AB — B, and ab=b 



.'. Wl 



-iirl^^'-'^} 



where W is the weight applied at the middle 
between the supports, and /is a constant depend- 
ing on the nature of the material. 



Fluid for Etching on Copper. — Yerdigris 4 parts ; salt 4 ; sal 
ammoniac 4; alum 1 ; water 16; strong vinegar 12. Dissolve with 
heat. 

Acid for Etching on Steel. — Pyroligneous acid 5 parts ; alcohol 
1 ; nitric acid 1. ISIix the first two, then add the nitric acid. 



Strain and Stress of Materials. 



215 



TABLE 

Of Experiments on the Transverse Strength of Rectangular Tubes of 
Wrought-IroUy supported at each end, and the weight laid on the 
middle. 



Distance between 
the supports. 



Weight of tubes 
between the sup- 
ports. 



Breaking 
weights, exclu- 
sive of the 
weights of the 
tubes. 


External 

depth of 

the 

tubes. 


External 
breadth 
of the 
tubes. 


Tons. 


Inches. 


Inches. 


57-5 


24 


16 


4-454 


6 


4 


22-84 


24 


16 


1-409 


6 


4 


1-1 


3 


2 


-3 


3 


2 


114-76 


36 


24 


1-1 


3 


2 


54-3 


24 


16 



Thickness of 

the plates of 

the tubes. 



Feet. 

30-0 
7-5 

30-0 
7-5 
3-75 
3-75 

45-0 
3-75 

30-0 



42-62 cwt. 
72-36 lbs. 
23-09 cwt. 
35-53 lbs. 

9-65 '' 

4-34 " 
130-36 cwt. 

9-65 lbs. 

39 cwt. 



Inches. 

•525 
-1325 

-272 

•0G5 

•061 

•03 

•75 

•061 

•50 



In several of these experiments the tubes gave way by the metal 
at the top becoming wrinkled. 

In similar tubes the strength, and consequently the breaking 
weight, is proportional to (1*9) power of the lineal dimensions. 

From these experiments the breaking weight may be obtained as 
follows : 



W-- 



hd^l 



in tons. 



The breadths and depths are measured in inches, and the length 
in feet. 

If the thickness of the metal be equal to t inches completely 
round the section, 



Then, 



.(B---2t){J)-'2ty 



the breaking weight in tons for a wrought-iron tube, 

whose form of section is 

What is tlie breaking weight of a rectangular tube 40 
feet long, depth 2 feet 6 inches, thickness of plate J- inch, 
and breadth 18 inclies? 






L 



216 



Strain and Stress of Materials. 



1600 ( 



486000-^449267 



: 22-96 tons. 



From a great number of well arranged experiments, on the strength 
of iron beams and tubes, it follows that they may be safely reduced 
in strength from the middle towards the extremities in the ratio 
indicated by theory. 



3> 



^K 



A 



Let AB he ^ beam supported at its extremities ^ and i^, and 
put i^ equal to the necessary strength at the middle of the beam. 

^l (ji Q jyi 

Then, the necessary strength at i> = i^ x — --— ^ 

The tensile force of wrought iron is to its compressive force as 2 
to 1. 

Hence, the plate on the upper side of hollow wrought-iron tubes 
should contain an area twice as great as the plate on the under 
side. 

Strength of Cast-iron Pillars. 
The breaking weight of solid cylindrical cast-iron pillars. 
In solid pillars, with their ends rounded, and moveable, 

Breaking weight in tons = 14*9 x — ... (1) 

In solid pillars, with their ends flat, and incapable of motion, 

Breaking weight m tons = 44*16 X — . . . (2)* 

where / is the length of pillar in feet, and d the diameter in inches. 

In hollow pillars of cast-iron, where D, d are the external, internal 

diameters, and I the length : both ends of the pillar were moveable. 



Breaking weight in tons = 13 x 



I'- 



In hollow cast-iron beams, whose ends were flat and firmly fixed, 

D^<^—d^'^ 
Breaking weight in tons r= 44*3 yvi 

Of three cylindrical pillars of steel, wrought and cast iron, and 
wood, all of the same length and diameter, the first having its ends 

* Formula (l) was obtained from the mean result of eighteen pillars, varying in length 
from 121 times the diameter down to 15 times. The formula (-2) was derived from eleven 
pillars, with flat ends, varying in length from 78 to 25 times the diameter. 



Strain and Stress of Materials. 21'i 



rounded, the second with one end round and the other end flat, and 
the third with both ends flat, the strengths are as 1, 2, and 3. 

These formula and results were obtained from experiments on 
pillars, varying in length from 121 times the diameter down to 15 
times. 

Effects of Temperature iipoii the Strength of Cad-Iron. 

The strength of cast-iron is not reduced when its temperature is 

raised to 600°, which is nearly that of melting lead ; and it does not 

diff'er very widely whatever the temperature may be, provided the , 

bar be not heated so as to be red hot. [ 

Example. 
Find the strength of a hollow cylindrical cast-iron pillar, 14 feet 
long, 6'2 inches external diameter, and 4*1 inches internal; the pil-; 
lar being flat and well supported at the ends. ^ 

14^-^ = 88-801 6-22'« = 712-22 and 4-1'-" = leO-'Z | 

.*. Breaking weight in tons = 44*3 x ■ 

712-22 -160-7 
= ""-^" 88-801 

= 275 
Comparative Strenc/th of Long Pillars. 
If the strength of cast-iron pillars be 1000, then wrought-iron 
will be 1745, cast-steel 2518, Dantzic oak 108*8, and red deal 78*5. 
The strength of similar pillars is as the square of their linear 
dimensions. 

Resistance to Torsion. 
Let I = length of prism from the fixed end to the point of appli- 
cation of the lever used to twist it 
r = radius of prism, if round. 
b, d=^ breadth and thickness, if rectangular. 
IK = the weight acting by means of the lever to twist the 

prism. 
L = length of the lever to which the weight TTis applied. 
6 = angle of torsion. 

B = resistance to torsion at the time of fracture. 
C = constant for each species of body. 

77 = 3-14159, (fee. 
For a cylinder, 

2 LI W= CnOr* iind 2 WL = 7t Rr. 

For a square, 

6 LI W= CO d* and 6 WL= i/2'Bd'. 

For a rectangle, 

ZLl W(b'' + d''):=:Cdb'd^ and 3 WL \/ b"" + d"" ^ R U' d!" 



218 Strain and Stress of Materials. 



The Ultimate Resistance of a Cast-iron Beam to Torsion, 
In a cylinder, TFX = 51055 r^ 
In a square prism, irX=7660 <P. 

tr d? 
In a rectanorular prism, W L == 10834 == — 

All the dimensions are taken in inches. 
Strength of Ropes. 

The cohesion of hempen fibres is 6400 lbs. for every square inch 
of section. 

-P, , . • 1. ^ • circumference squared 

JDreakinff weisrnt in tons = ^=— 

the circumference being measured in inches, 

Ex. — Find the breaking weight of a rope G inches in circum- 
ference. 

Breaking weight =: — = 9 tons. 

For a eommon cable^ 

^ , . . , ^ . ^ circumference squared 

Breaking weight m tons = ~ 2 

5 

These are practical rules and easy of application. 



PEOCESSES FOE STAmiKG WOODS. 

Mahogaiiy Color {Dark). — Boil ^ lb. of madder and 2 oz. of log- 
wood in a gallon of water; then brush the wood well orer with 
the hat liquid. When dry, go over the whole with a solution of 2 
drachms of pearlash in a "quart of water. 

Mahogany Color {Light). — Brush over the surface with diluted 
nitrous acid, and when dry apply the following, with a soft brush: 
Dragon's blood, 4 oz. ; common soda, 1 oz, ; spirit of wine, 3 pints. 
Let it stand in a warm place, shake it frequently, and then strain. 
Repeat the application until the proper color is obtained. 

To Stain Maple a Mahogany Color. — Dragon's blood, ^ oz. ; 
;alkanet, i oz. ; aloes, 1 dr. ; spirit of wine, 16 oz. Apply it with a 
.-^ponge or brush- 

Rosewood. — Boil 8 oz. of logwood in 3 pints of water until re- 1 
dueedto half; apply it, boiling hot, two or three times, letting it dry i 
between each. Afterwards put in the streaks, with a camel's hair 
j)encil, dipped in a solution of copperas ;a^d yerdigris in a decoction 
oi logwood. 



Logarithms. 



219 



i 



Ebony. — Wash the wood repeatedly with a solution of sulphate 
of iron ; let it dry, then apply a hot decoction of logYv^ood and nut- 
galls for two or three times. When dry, wipe it with a wet sponge; 
and when dry again, polish with linseed oil. 

Red. — 1. Take a pound of Brazil wood and mix it with a gallon 
of stale urine. Pour over the wood while boiling hot. Before it 
dries it should be laid over with alum water. 2. A fine red may also 
be obtained by a solution of dragon's blood in spirits of wine. 

Yellow. — Xilric acid, lightly diluted, will produce a fine yellow 
on wood. Sometimes, if the wood is not in proper condition, it will 
create a brown. Care must be taken that the acid used be not too 
strong, or it will render the wood nearly black. 

Blue. — Take of alum 4 parts ; water 85 parts. Boil. 
Purple. — To produce this color, take of logwood 11 parts; alum 
3 parts ; water 29 parts. Boil. 

Mahogany. — 1. Linseed oil 2 pounds; alkanet 3 ounces. Heat 
them together and macerate for six hours, then add resin 2 ounces; 
beeswax 2 ounces. Boiled oil may be advantageously used instead 
of the linseed oil. 

2. Brazil-wood (ground) ; water sufficient ; add a little alum and 
potash. Boil. 

3, Logwood 1 part ; water 8 parts. Make a decoction, and apply 
it to the wood; when dry, give it two or three coats of the follow- 
ing varnish: dragon's blood 1 part; spirits of wine 20 parts. Mix. 

To take Stabis out of Mahogany. — Spirits of salts 6 ]:arts; salt 
of lemons 1 part. Mix, then drop a little on the stains, and rub 
them until they disappear. 

To Stain Musical Instruments. — Crimson : Boil one pound of 
ground Brazil Avood in three quarts of w^ater for an hour; strain it, 
and add half an ounce of cochineal ; boil it again for half an hour 
gently, and it will be fit for use. 

Purple : Boil a pound of chip logwood in three quarts of water 
for an hour ; then add four ounces of alum. 



LOGARITHMS. 

Logarithms literally signify ratios of numbers ; hence Logarithmic 
Tables may be various, but those in common use for the fjicilitating 
of arithmetical operations generally are of the following corres- 
ponding progressions, viz. : — 

Arithmetical, 0, I, 2. 3. &c., or series of loffarilhms. 
Geometrical, 1. 10, 100, 1000, &c., or ratio ofnumbers. 

And thus it may be perceived, that if the log of 10 be 1, the 

log. of any number less than 10 must consist wholly of decimals, 

because increasing by a decimal ratio. Again; if the log. of 100 



•220 



Logarithms. 



I be 2, the log. of any intermediate number between 10 and 100 
I must be 1, with so many decimals annexed ; and in like manner, 
the log. of any intermediate number between 100 and 1000, must' 
be 2, with decimals annexed proportionally, as before. ! 

Application and Utility of Common Logarithmic Tables, f 

The whole numbers of the series of logarithms, as 1, 2, 8, <fee."; 
are called the indices, or characteristics of the logarithm, and whieli 
must be added to the logarithm obtained by the table, in propor- 
tion to the number of figures contained in the given sum. Thus 
suppose the logarithm be required for a sum of only two figures^ 
the index is 1 ; if of three figures, the index is 2 ; and if of four 
figures, the index is 3, (fcc. ; being always a number less by unity 
than the number of figures the given sum contains. 

Examples. 

The index of 8 is 0, because it is less than 10. 

The index of 80 is 1, because it is less than 100. 

The index of 800 is 2. because it is less than 1000. 

The index of 8000 is 3, because it is less than 10,000, kcu 

The index of a decimal is always the number which denotes the 
significant figure from the decimal point, and is marked with the 
sign, thus, — , to distinguish it from a whole number 

ExA^IPLi^S. 

The index of •325-49 is — 1, because the first significant figure is 
the first decimal. 

The index of -082549 is — 2, because the first significant figure is I 
the second decimal. 

The index of -0032549 is — 3, because the first significant figure 
is the third decimal, (fcc, of any other sum. 

If the given sum for which the logarithm is required contaios or 
consists of both integei^ and decimals, the index is determined by 
the integer part, without having any regard to the other. 

1. To find the logarithm of anxj ichole nv.mher under 100. 

Look for the number under X in the first page of any Logarithmic 
Table; then immediately on the ridit of^it is the Toorarithm 
required, with its proper index. Thus the loe of 64 is 1-806180, 
and the log. of ^72 is 1-857332. " j 

2 To find th^ logarithm of any number between 100 and 1000, cA 
any sum not exceeding 4 figures. ' | 

Find the first three figures in the left-hand column of the page 
under X. in which the number is situated, and the fourth figure, at 
the top or bottom of the page ; then the logarithm directly under 
the fourth figure, and in a line with the three figures in the column 
on the left, with its proper index, is the loirarithm required. Thus, 
the log. of 450 is 2 653213, and the locr. of 7464 is 3-872972. Or, 
the log. of 378-5 is 2-578066, and that of -7854 is — 1-895091, 



Logarithms. 221 



3. To find the number indicated by a given logarithm. 

Look for the decimal part of the given logarithm in the different 
columns, and if it cannot be found exactly, take the next less. Then 
under IST in the left-hand column, and in a line with the logarithm 
found, are three figures of the number required, and on the top of 
the column in which the found logarithm stands is one figure more ; 
place the decimal point as indicated by the logarithmic index, 
Avhich determines the sum, properly valued, as required. 

If the logarithm cannot be found exactly in the tables, subtract 
from it the next less that can be found, and divide the remainder by 
the tabular difference ; the quotient will be the rest of the figures 
of the given number, which, being annexed to the tabular number 
already found, is the proper number required. 

Ex. Required, the number answering to the logarithm 3'233568. 
Given logarithm . . . . =n 3*233568 
Next less is the log. of 1112=: 3-233504 



Remainder . , . . 64 

64 
Tab. Difi". = 253, and— - = 2-5 
253 

Hence the number required = 1'712'25. 

For practical purposes in mechanics, logarithms are seldom 
resorted to, unless for the raising of the powers of numbers or 
extraction of their roots. These operations, when tables are at 
hand, they very much facilitate ; involution or the raising of powers, 
being performed simply by multiplication, and evolution, or the 
extraction of roots, by division, as in simple arithmetic. 

Ex. 1. Required, the square or second power of 25-79L 

Log. of 25-791 = 1-411468 

Multiplied by 2 the power required. 

Logarithm 2*822936 indicated number or square required 
r= 665-175. 

Ex. 2. What is the cube of 30-7146 ? 
Logarithm - 1 '487 34 5 
Multiplied by 3 the power required. 

Logarithm 4-462035 indicated number or cube required 

= 28975-7. 

Ex. 3. Required, the square root of 365. 

Log. = = 1-281146 indicated number or root = 19-105. 



19* 



222 Logarithms. 



Ex. 4. Find the cube root of 12345. 

4*091491 
Log.= ^— ^— = 1-863830 indicated number or root = 23-116. 



For T.ABLE OF Logarithms, see p, 48 3» 



Engraving in Alto-Relievo. — In the common operation of engra- 
ving, the desired effect is produced by making incisions upon the 
copper-plate with a steel instrument of an angular shape, which 
incisions are filled with printing-ink, and transferred to the paper 
by means of a roller, which is passed over its surface. There is 
another mode of producing these lines or incisions, by means of 
diluted nitrous acid, in which the impression is taken in the same 
way. Another method of engraving is done upon a principle 
exactly the reverse, for instead of the subject being cut into the 
copper, it is the interstice between the lines which is removed by 
diluted aquafortis, and the lines are left as the surface, from which 
the impression is taken by means of a common type-printing press, 
instead of a copper-plate press. 

This is effected by drawing with common turpentine varnish, 
covered with lampblack, whatever is required upon the plate; and 
when the varnish is thoroughly dry, the acid is poured upon it, and 
the interstice of course removed by its action upon the uncovered 
part of the copper. If the subject is very full of dark shadows, 
this operation will be performed with little risk of accident, and 
with the removal of very little of the interstice between the lines; 
but if the distance between the lines is great, the risk and difficulty 
is ver}' much increased, and it will be requisite to cut away the 
parts which surround the lines with a graver, in order to prevent 
the dabber with the printing-ink from reaching the bottom, and 
thus producing a blurred impression. It is obvious, therefore, that 
the more the plate is covered with work, the less risk there will be 
in the preparation of it with the acid, after the subject is drawn, 
and the less trouble will there be in removing the interstice, if any, 
from those places where there is little shading. 

Glass, Soluble. — Mix ten parts of carbonate of potash, fifteen of 
quartz (or of sand free from iron of alumina), and one part of char- 
coal. Fuse together. The mass is soluble in four or five parts of 
water; and the filtered solution evaporated to dryness yields a 
transparent glass, permanent in the air. 



Movable Points i\ a Parallel Motion. 



223 



TABLE 

By which to Determine the variofs Distances of the Movable Points 
in a Parallel 3fotion. 



Q 


•S 


J-S 


s 


_c 


«-3 

• i C 


S 


•S 


.1 = 


e 


_C 


=■§ 


CS 




-a a 






-3 ;-J 


== 




^ H 






r:; :3 


<U*J 




CO *J oj 


2^*-' 


2 


« •" ffl 




2: 


2 ■" M 


b^ 


i2 


==■" r^ 


Xi (U 


c3 • 


■- (U d 


.X2 D 


a • 


~ <U OJ 


j3 « 


«^ 




J2 <vi 


C3*^ 


t, d) w 


11 


— <2J 




■p" 








"1 


:li 


0=^ 




1=1 


^ 

s 






1 




fl 


'3 


■3 


P- 


■5 


5 


|1 
6 Oi 




2 


2 




3 


4 H 


I 


3f 


6 Oi 




2i 


1 4f 




Si- 


3 3 




4 


5 6f 




5i 


5 3 




2f 


lOi 
6i 




Si 
Sf 


2 6i 

2 01 




4i 
44 


4 3 

3 61 





54 
5i 


4 6i 
3 11 




3 


4 


r^:. 

«© 


4 

4i 


1 6f 

1 H 




4t 


2 IH 


0^ 


6 


3 31 










QO 


5 


2 51 


'"' 


6-i: 


2 lOf 




2 


3 H 




4i 


lOi 




5i 


2 Oi 




64 


2 of 




2J 


2 3 










54 


1 H 




6| 

5i 


2 1 




2i 

2f 
3 


1 n 

1 If 
9 

5| 




3 


5 4 
4 4 









4 


6 3 


6 3^ 




3i 




H 


3 6 




4i 


5 3f 




54 


5 6 






3i 
4 


3 2| 
2 3 


^ 

q; 


44 
4f 


4 6 
3 91- 




51 
6 


4 9i 
4 2 




2 

H 
H 

2f 
3 


4 6 
3 4f 
2 6 
1 lOi 
1 4 




x^- 


4i 


1 9 


^ 


5 


3 2^ 


rH 
1—1 


6i 


3 7f 






44 
4t 
5 


1 4| 
1 0-J 
9f 


Oi 


5i 
54 
5f 
6 


2 8i 
2 2f 
1 10 
1 6 




64 
6f 


3 If 

2 8i 






6 &i 




3i 


Hi 












54 




H 


7f 




3^ 


4 Qi 










5f 


9 










3f 


3 9 




44 


5 6f 


^ 


6 


5 Oi 




H 


4 8f 


0) 


4 


3 0| 




4f 


4 9 




6i 


4 5 




H 


3 7i 


^ 


4i 


2 6 




5 


4 Of 


r-p. 


64 


3 lOi 


q> 


2f 


2 9 


r^^ 


44 


2 


^ 


5i 


3 5| 


1— 1 


6f 


3 4 


^ 


3 


2 1 


^ 


4f 


1 7 




54 


2 10| 




7 


2 lOf 




H 


1 f)f 




5 


1 3 




5f 


2 6i 




n 


2 f> 




H 


1 If 

9i 




H 


IH 




6 


2 Oi 












6 8i 




3f 
















5f 










H 


5 9^ 




4i 


5 H 




6 


6 






1 




2f 


3 10 ! 




3f 


4 9f 




5 


5 




6i 


5 3-^ 




3 


3 i 




4 


4 




5J 


4 3i 


•4i 


64 


4 n 




H 


2 3i, 




4i 


3 3f 


0) 


H 


3 8i 


^ 


6f 


4 1 


<^ 


H 


1 9^ 


^ 


H 


2 8f 





5| 


3 If 


rH 


7 


3 H 


CO 


n 


1 4i 


CO 


4f 


2 21 




6 


2 8 




H 


3 If 




4 


1 




5 


I n 




6i 


2 3 




n 


2 8i 




4i 


SI 




5i 


1 5i 




6i 


1 UH 




7f 


2 4 





224 



Capillary Attpactiox, 



CAPILLAET ATTEACTION. 

If a number of glass tubes, open at both ends, be immersed, the 
water will rise to the same height in each tube, so long as the 
diameter of the tube exceeds the lifteenth of an inch ; in all tubes 
less than this, the water will rise higher in the tube whose diameter 
is the least. Such tubes, whose diameters are less than one fifteenth 
of an inch, are called capillary tubes, from the Latin word capillus, 
signifying a hair. 

Phenomena of Capillary Attraction. 
Let P Q RS\)Q 2i vessel containing water to the line P S. 

water will rise 



"iXu 


^=J:/ 



A 



The 

in the 
capillary tubes ^^C to 
tli£ heights mno, which 
are inversely proportional 
to their diameter. If B be 
broken at a, the water 
will not rise to the top of 
it, but will stand at b, a 
little below the top, what- 
" ever be the length or 

diameter of the tube. And, if the tube be taken out of the water 
and laid horizontally, the water will recede from the end that was 
immersed. 

Jf a tube D be composed of two different bores, the water willj 
rise to the height p; and if another tube, E, of the same form andj 
size, be immersed, with its smaller end downwards, the water will | 
rise in it to the same height p. j 

If the vessel Fvvj be plunged into water, and by exhaustion the? 
water is raised to the capillary tube Ftv;, it will afterwards ascend j 
to the height r, which is just the same as in a capillary tube G of, 
the same bore as Flu, and length Fx. 

In tubes of the same matter, immersed in the same fluid, the 
product of the elevations by the diameter is a constant quantity. 

In a glass tube, immersed in water, this constant has been found 
by Muschenbrock, -039; by AVeitbrecht, -0428; by Monge, '042; 
by Atwood -058. 

From these numbers, the diameter of a tube may be found, in 
which the water will rise, by capillary attraction, the height 7 
inches. 



Diameter : 



•039 



= '0056 inches, nearly. 



The constant quantity, here referred to, is called the modulus of 

capillary attraction. i 

The following m^oduli are from Brewster ; they were obtained ; 



Capillary Attraction. 



225 



with a glass tube of '0561 of an inch diameter, by means of an 
improved apparatus : 



Name of Fluid. 



Water, 

Very hot water, . . 
Muriatic acid, . . . 
Oil of boxwood, » . 
Oil of cassia, . . 
Mtrous acid, . o . 
Oil of rapeseed, . . 
Castor oil, .... 
Mtric acid, .... 
Oil of spermaceti, . 
Oil of almoads, . . , 
Oil of olives, . . . 
Balsam of Peru, . . 
Muriate of antimony, 
Oil of rhodium, . . 
Oil of pimento, . . 
Cajeput oil, .... 
Balsam of capivi, . . 
Oil of thyme, . . . 
Oil of bricks, distilled ) 
from spermaceti oil, j 
Oil of caraway seeds, 
Oil of rue, .... 
Oil of spearmn*:, . . 
Balsam of sulphur. 
Oil of sweet fennel ) 



Modulus. 



•0327 
•0301 
•0248 
•0240 
•0236 
•0232 
•0227 
•0226 
•0222 
•0220 
•0217 
•0215 
•0212 
•0209 
•0205 
•0203 
•0200 
•0200 
•0199 

•0199 

•0198 
•0198 
•0197 
•0196 

•0195 



Name of Fluid. 



Oil of h3'ssop, . . 
Oil of rosemary, . 
Oil of bergamot, . 
Oil of amber, . . 
Oil of anise seeds, . 
Oil of Barbadoes tar 
Laudanum, . . . 
Oil of cloves, . . 
Oil of turpentine, . 
Oil of lemon, . . 
Oil of lavender, 
Oil of camomile, . 
Oil of peppermint. 
Oil of sassafras. 
Highland whisk}^, . 
Brandy, .... 
Oil of wormwood, . 
Oil of dill seed, . . 
Oil of ambergris, . 
Oil of juniper, . . 
Oil of nutmeg, . . 
Alcohol, .... 
Oil of savin e, . . 

Ether, 

Oil of wine, . . . 
Sulphuric acid, . . 



Modulus. 



•0195 
•0193 
•0192 
•0192 
•0192 
•0191 
•0191 
•0187 
•0187 
•0187 
•0184 
•0184 
•0184 
•0184 
•0184 
•0183 
•0183 
•0182 
•0181 
•0180 
•0180 
•0178 
•0174 
•0160 
•0153 
•0112 



These experiments were made with a tube, carefully cleaned and 
dried after each experiment. A dry tube will raise the water to a 
les?i height than a wet one. 

When capillary tubes are plunged into mercury, it falls instead 
of rising, as is the case with other fluids ; and its fall is such, that 
when it is multiplied l)y the diameter of the tube, the product is a 
constant quantity 015 (Cavendish). 

When water is made to pass through a capillary tube of such a 
bore that the fluid is discharged only by successive drops, the tube, 
when electrified, will furnish a constant and accelerated stream ; 
and the acceleration is proportional to the smallness of the bore. 
A jet of warm water will rise to a much greater height than a jet 
of cold water, though the water in both cases moved through the 



226 



Woods. 



same aperture, and was influenced by tlie same pressure. A syphon 
wliich discharges cold water only b}- drops, will furnish warm 
water in an uninterrupted stream. 

Let G EE B, A DEB, be two plates of glass, having their sides 
EB joined together with wax, and 
their surfaces smooth and c^ean ; and 
also their sides, A B, C E, separated 
slightly so as to form the angle ABC. 
If this apparatus be plunged in a 
vessel, so that IHG represent the 
water's surface, then the water will 
rise between the plates of glass, by 
capillary attraction, to the height ' 
IE G, so that the boundary of the 
water on the planes FEBC, DEB A, \ 
will be the hyperbolas G E and IE, \ 
having for their asymptotes the sur- 1 
face of the fluid and the line EH. \ 
The height, n m, to which the water i 
will rise, is regulated entirely by the j 
same laws which prevail in the case 
of the tubes ; calling the distance, n o, j 
between the plates the diameter of. 
the tube. 

Hence the height, n m, is equal to ! 
the height in a tube whose diameter 
is equal to n o\ and so on for any other point. 

All phenomena of capillary attraction are exhibited equally both 
in air and in vacuo, and they are entirely independent of the thick- 
ness of the material composing the tubes and plates. 

The elevation and depression is not proportional to the density 
of the liquid; water stands much higher in a glass tube than 
alcohoL 




WOODS. 

How to Polish Wood. 
Take a piece of pumice-stone and water, and pass repeatedly 
over the work until the rising of the grain is cut down. Then take 
powdered tripoli and boiled linseed oil, and polish the work to a 
bright surface. 

To Gather a7id Preserve Woods. 
Woods should be gathered and exposed in a dry situation, to a , 
heat of from 90° to 150° Fah., until sufficiently dry. The larger' 
kinds are more easily chipped before drying. , j 



Steam-Engine. 227 



To Preserve Woodwoi'k. 
Take boiled oil and finely powdered charcoal; mix to the con- 
sistence of paint, and give the woodwork two or three coats with 
it. This composition is well adapted for casks, water-spouts, <fcc. 

To produce Figures on Wood. 
Slack some lime in stale wine. Dip a brush in it, and form on 
the wood figures to suit your fancy. When dry, rub it well with a 
rind of pork. 



STEAM-ENGINE. 

To Estimate^ hy means of an Indicator ^ the Amoiont of Effective 

Power produced by a Stea7n- Engine. 

Rule. Multiply the area of the piston in square inches by the 

average force of the steam in lbs., and by the velocity of the piston 

in feet per minute ; divide the product by 33,000, and ■^q^\\?> of the 

quotient equal the effective power. 

Ex. Suppose an engine with a cylinder of 374- inches diameter, 
a stroke of 7 feet, and making 17 revolutions per minute, or 238 
feet velocity, and the average indicated pressure of the steam 
16*73 lbs. per square inch; required the effective power. 

Area — 1104 '4687 inches x 16-73 lbs., x 238 feet. 
33000 
133-26 X 7 



10 



93 '282 horse power. 



To determine the proper Velocity for the Piston of a Steam- Engine. 

Ride. Multiply the logarithm of the ni\i part of the stroke at 
which the steam is cut off by 2*3, and to the product of this add 
•7. Multiply the sum by the distance in feet the piston has tra- 
velled when the steam is cut off, and 120 times the square root of 
the product will equal the proper velocity for the piston in feet 
per minute. 

Ex. Let the steam be cut off in an 8-feet stroke when the piston 
has travelled Jth of the length j required its proper velocity. 

Logarithm of 4 = 0-60206 



Multiplied by 
To which add 


2-3 

1-384738 

•7 




2-084788 
2 



♦/4-169476 =r 2-04 x 120 == 245 feet, velocity per 
minute. 



228 



Properties and Effects of Heat. 



TABLE 

Of Approximate Velocities for the Pistons of Steam- Engines. 



CONDENSING ENGINES. 


' NON-CONDENSING ENGINES. 


Length 




Number of 


Length 




Number of 


of stroke 


Velocity in feet 


revolutions per 


of stroke in 


Velocity in 


revolutions 


in feet 


per minute. 


minute. 


feet. 


feet per minute. 


per minute. 


2 


160 


40 


1 H 


186 


62 


2i 


lV7i 


354 


2 


200 


50 


8 


192 


32 


i 24 


2124 


424 


H 


203 


29 


2i 


2l7i 


394 


4 


214 


26f 


1 3 


222 


37 


^ 


220i 


244 


1 34 


231 


33 


6 


230 


23 


i 4 


236 


294 


54 


2364 


214 


44 


243 


27 


6 


240 


20 


5 


2474 


24f 


Y 


245 


174 


54 


253 


23 


8 


256 


16 


6 


264 


22 



Of the Parallel Motion in a Steam-Engine. 
When the power from the piston is communicated by means of a 
beam or lever moving upon an axis, the parallel motion becomes a 
very important portion of the machine ; for then it forms the link 
of connexion, and by its properties renders the action of alternate I 
circular motion, and reciprocating vertical motion, mutually agree- 
able, thereby properly insuring to the piston-rod a truly direct line 
to that of the cylinder; but. to effect this, the greatest degree of ! 
exactitude of the various parts is required, otherwise extra friction 
is created, and the effective power of the engine proportionately 
diminished. 



THE PKOPERTIES K^V> MISCELLANEOUS 
EFFECTS OF HEAT. 



Linear 


Expansion of Metals 


from 32° to 212°.--Faraday. 


Zinc, 1 


part in . 


. 322 


Gold, I part in . . 682 


Lead, 


" 


. 351 


Bismuth, 


. 719 


Tin, pure, 


c« 


. 403 


Iron, " 


. 852 


Tin, impure, 


l< 


. 500 


Antimony, " 


. 923 


Silver, 


(( 


. . 524 


Palladium, " 


. 1000 


Copper, 


ii 


. . 581 


Platinum, " 


. 1100 


Brass, 


u 


. . 684 


Flint Glass, " 


. 1248 



Properties and Effects of Heat, 



229 



TABLE 

Of the Expansion of Water by Heat. — By Dalton. 



Temperatute. 


Expansion. 


Temperature. 


Expansion. 


12° Fahrenheit. 


100236 


122° Fahrenheit. 


101116 


22 , 


100090 


132 


101367 


32 


100022 


142 


101638 


42 


100000 


152 


101934 


62 


100021 


162 


102245 


62 


100083 


172 


102575 


n 


100180 


182 


102916 


82 


100312 


192 


103265 


^2 


1004^7 


202 


103634 


102 


100672 


212 


104012 


112 


100880 







TABLE 

Of the Heating Power of various Combustible Substances, exhibiting 
the utmost quantity of Water evaporated by the given Weights, and 
the smallest quantity of Air capable of producing total Combustion. 
Dr. Ure. 



Species of Combustible. 



Pounds of water 
which a pound I Pounds of boil- 
can heat from 0°|ing water evapo- 
to 212*^. Tated by I pound. 



Weight of at- 
mospheric air at 
32" to burn 1 
pound. 



Perfectly dry wood, . . 
Wood in its ordinary state, 
Wood charcoal, .... 

Pit coal, 

Coke, 

Turf, 

Turf charcoal, 

Carburetted hydrogen gas, 

Oil, 

Wax, 

Tallow, 

Alcohol of the shops, . . 



35-00 
26-00 
73-00 
60-00 
65-00 
30-00 
6400 
76-00 

78-00 

52-60 



20 



4-72 
13-27 
10-90 
11-81 

5-45 
11-63 
13-81 

14-18 

9-56 



Smallest quantity. 

5-96 

4-47 
11-46 

9-26 
11-46 

4-60 

9-86 
14-58 

15-00 

11-60 



280 



Properties and Effects of Heat. 



TABLE 

Of boiling points of water holding various proportions of salt in 

solutio7i. 





Parts of 


Decrees of 


Degrees of 


Degrees of 




Salt. 


Fahrenheit. 


Reauiner. 


Centigrade. 


Saturated solution . . . 


se'SY 


226-6 


86-2 


107-8 


a 


(( 








33-34 


224-9 


85-7 


107-2 


" 


a 








30-30 


223-7 


85-2 


106-5 


(( 


it 








27-28 


222-5 


84-7 


1058 


« 


n 








24-25 


221-4 


84-1 


105-2 


tt 


a 








21-22 


220-2 


83-6 


1046 


ei 


it 








18-18 


219 


83 


103-9 


u 


(t 








15-15 


217-9 


82-6 


103-3 


a 


li 








12-12 


216-7 


82-1 


1026 










9-09 


215-5 


81 6 


102 


a u 








6 06 


214-4 


81-1 


101-3 


Sea-water . . . 








8-03 


213-2 


80-5 


100-7 


Commoii water . 








0-00 


212 


80 


100 



Expansion of Liquids in Volume from 32° to 212° Fahrenheit 
1000 parts of water become 1046 



oil 
mercury 
spirits of wine 



air 



1080 
1018 
1110 
1373 



Of the Linear Dilatation of Solids hy Heat 
a bar takes at 212^, ichose length at 32 



Cast iron, . . . 1-00111111 
Steel (rod), . . . 1-00114470 
Steel, not tem- ) 

pered, . . . ) 
Ditto, temper- ) 

ed yellow, . \ 
Ditto, at a high- \ 

er rate, ... J 

Iron, 1-00118203 

Soft iron, forged, . 1-00122045 

Gold, 100150000 

Copper, . . . . 1-00191000 



Cast brass, 



1-00107875 
1-00136900 
1-00123956 



Dimensions which 
^s 1-000000. 

. . . 1-0018750 

Silver, 1-0018900 

I Tin, 1-0028400 

j Lead, 1-00284836 

Zinc, 1-00294200 

I Glass from 32° ) 

! to 212°, . . f 

Glass from 212° ) 

to 392°, . . \ 

Glass from 392' " 

to 572°, 



92°) 



1-00086130 
1-00091827 
1-00101114 



I 



Expansion of Atmospheric Air. 



231 



Of Capacities of Bodies for Heat referred to Water as the Standard. 

1-0000 Iron, -1300 

. -7100 Hardened steel, . . . . -1230 

. -5280 Steel softened by fire, . -1200 

. -4720 Soft bar iron, . . . . -1190 

. -0330 Brass, -1160 

. -9000 Copper, -1140 

. -2777 Zinc, . -1000 

. -2700 Ashes of charcoal, . . . -0909 

. =2300 Silver, -0520 

. -1900 Tin, -0704 

. -1855 White lead, -0670 

. -1830 Gold, -0500 

. -1402 Lead, -0420 



Water, .... 
Olive oil, . . . 
Linseed oil, . . 
Oil of turpentine, 
Quicksilver, . . 
Ice, 



Pit coal 

Chalk, .' . . . . 
Sea salt, . . . . 
Sulphur, . . . . 
Ashes of cinders. 
Black lead, . . . 
Ashes of elm wood, 



TABLE ^ 

Of the Expansion of Atmospheric Air hy Heat 



Degrees of 




Dejrrees of 




Degrees of 




Fahrenheit. 


Bulk. 


Fahrenheit. 


Bulk. 


Fahrenheit. 


Bulk. 


32° 


1000 


65° 


1077 


100° 


1152 


35 


1007 


70 


1089 


120 


1194 


40 


1021 


75 


1099 


140 


1235 


45 


1032 


80 


1110 


160 


1275 


50 


1043 


85 


1121 


180 


1315 


55 


1055 


90 


1132 


200 


1364 


60 


1066 


95 


1142 


212 


1376 



The pressure or gravity of the atmosphere, being equal to a 
column of water 34 feet in height, is the means or pi-inciple on 
which rests the utility of the common pump, also of the s^'phon, 
and all other such hydraulic applications. In the pump, the internal 
pressure on the surface of the liquid is removed by the action of the 
bucket; and as by degrees the density becomes lessened, so the 
water rises by the external pressure to the above-named height ; 
and at such height it will remain, unless by some derangement of 
construction taking place, the atmospheric fluid is allowed to enter 
and displace the liquid column. But observe, if the temj^erature 
of the water or other liquid be so elevated that steam or vapor 
arise through it, then, according to the vapor's accumulation of 
density, may the action of the pump be partially or Avliolly 
destroyed ; and the only means of evasion in such cases is to place 
the working bucket beneath the surface of the liquid which is 
required to be raised. 





232 


Degrees of the Three Thermometrical Scales. 










TABLE 






Of the Degrees of the three Thermometrical Scales, 






Above Boilin,^ Point of Water. 




g« 


•z ^ 


3 u. 


aJ*J 


■■z6 


3 C 


l^ 


•i« 


i C 


S^ 


•i;_a5 


3 C 




•h <o 


3 c 




1^ 


§1 




1^ 


11 










'& 


«1 
U be 


11 


J3 2 

248 


%1 
120 


-1 
96 




392 


200 


160 


356 


180 


144 


320 


160 


128 


284 


140 


112 




391 






355 






319 






283 






247 










390 


199 




354 


179 




318 


159 




282 


139 




246 


119 








389 
388 


198 


159 


353 

352 


178 


143 


317 
316 


158 


127 


281 
280 


138 


111 


245 

244 


118 


95 






387 
386 


197 


158 


351 
350 


177 


142 


315 
314 


157 


126 


279 

278 


137 


110 


243 
242 


117 


94 






385 


196 


157 


349 


176 


141 


313 


156 


125 


277 


136 


109 


241 


116 


93 






384 






348 






312 






276 






240 










383 


195 


156 


347 


175 


140 


311 


155 


124 


275 


135 


108 


239 


115 


92 






382 






346 






310 






274 






238 










381 


194 




345 


174 




309 


154 




273 


134 




237 


114 








380 




155 


344 




139 


308 




123 


272 




107 


236 




91 






379 


193 




343 


173 




307 


153 




271 


133 




235 


113 








378 




154 


342 




138 


306 




122 


270 




106 


234 




90 






377 


192 




341 


172 




305 


152 




269 


132 




233 


112 








376 


191 


153 


340 


171 


137 


304 


151 


121 


268 


131 


105 


232 


111 


89 






375 






339 






303 






267 






231 










374 


190 


152 


338 


170 


136 


302 


150 


120 


266 


130 


104 


230 


110 


88 






373 






3.37 






301 






265 






229 










372 


189 




336 


169 




300 


149 




264 


129 




228 


109 








371 




151 


335 




135 


299 




119 


263 




103 


227 




S7 






370 


188 




334 


168 




298 


148 




262 


128 




226 


108 








369 




150 


333 




134 


297 




118 


261 




102 


225 




86 






368 


187 




332 


167 




296 


147 




260 


127 




224 


107 








367 


186 


149 


331 


166 


133 


fj". 146 


117 


259 


126 


101 


223 


106 


85 






366 






330 






294 




258 






222 










365 


185 


148 


329 


165 


132 


293 145 


116 


257 


125 


100 


221 


105 


84 






364 






328 






292 1 




256 






220 










363 


184 




327 


164 




291 


144 




255 


124 




219 


104 








362 




147 


326 




131 


290 




115 


254 




99 


218 




83 






361 


183 




325 


163 




289 


143 




253 


123 




217 


103 








360 




146 


324 




130 


288 




114 


252 




98 


216 




82 






359 

358 


182 


145 


323 
322 


162 


129 


287 
286 


142 


113 


251 

250 


122 


97 


215 
1 214 


102 


81 






357 


181 




321 


161 




285 


141 




249 


121 




213 

1 


101 






To convert the Degrees in the three Scales into each other. 




To convert Centigrade or Reaumur's into Fahrenheit's Degrees— Multiply the 






number of degrees by 9, divide the product by 5 tor Centisrade. or^by 4 for Reaumur's ; 






add 32 to the quotient, and the sum will be degrees of Fahrenheit. 






To convert Fahrenheit's into Centigrade or Rea,umur's Degrees.—BnhtrAci 32 from 






the number of degrees, and divide the remainder by 9; multiply the quotient by 5 for 






Centigrade, or 4 for Reaumur's ; the products will be the required degrees respectively. 















Deg 


REES 


OF THE Three Thermometrtcal Scales. 




233 


Comparative Table of the Degrees of the three Thermoineirical Scales, 


Fahr't 


Cent. 


Rea. 


JFahr't 


Cent. 


Ren. 


P:ilir't 


Cent. 


Rea.;i Fahr't 


Cent 


Rea.j I Fahr't 


Cent. 


Re a. 


212 


100 


80 


167 


75 


60 


122 


50 


40 


7Y 


25 


20 


! 32 








211 






166 






121 






76 






1 31 


( 




210 


99 


79 


165 


74 


59 


120 


49 


39 


75 


24 


19 


i 30 


1- 1 


- 1 


209 






164 






119 






74 


; 


; 29 




208 
207 


98 
97 


78 


163 
162 


73 

72 


58 


lis 

117 


48 

47 


38 


73 

72 


23 
22 


18 


1 28 

i 27 


- 2 

- 3 


- 2 


206 






161 






116 






71 






1 26 






205 


96 


77 


160 


71 


57 


115 


46 


37 


70 


21 


17 


I 25 


- 4 


- 3 


204 






159 






114 






69 






i 24 






203 


95 


<6 


158 


70 


56 


113 


45 


36 


68 


20 


16 


23 


- 5 


- 4 


202 






157 






112 






67 






22 






201 


94 


75 ^ 


156 


69 


55 


111 


44 


35 


66 


19 


15 


21 


- 6 


- 5 


200 






155 






110 






65 






20 




199 
198 


93 

92 


H 


154 
153 


68 
67 


54 


109 

108 


43 

42 


34 


64 
63 


18 
17 


14 


: 19 

1 ^l 


- 7 

- 8 


- 6 


197 






152 






107 






62 






17 




- 7 


196 


91 


73 


151 


66 


53 


106 


41 


33 


61 


16 


13 


16 


- 9 


195 




! 


150 






105 






60 






15 






194 


90 


72 ; 


149 


65 


52 


104 


40 


32 


59 


15 


12 


14 


-10 


- 8 


193 






148 






103 






68 






13 






192 


89 


71 ! 


147 


64 


51 


102 


39 


31 


57 


14 


11 


12 


-11 


- 9 


191 




1 


146 






101 






56 






11 


-. ^1 


190 


88 


' 


145 


63 




100 


38 




55 


13 




10 


-12 




189 


87 


70 


144 


62 


50 


99 


37 


30^ 


54 


12 


10 


9 


-13 


-10 


188 




U3 






98 






53 






8 






187 


86 


69 


142 


61 


49 


97 


86 


29 


52 


11 


9 


7 


-14 


-11 


186 






141 






96 






51 






6 






185 


85 


68 


140 


60 


48 


95 


35 


28 ' 


50 


10 


8 


5 


-15 


12 


184 






139 






94 






49 






4 




1 


183 


84 


67 


138 


59 


47 


93 


34 


27 I 


48 


9 


7 


3 


-16 


1 


182 






137 






92 






47 




2 




-13 


181 


83 




136 


58 




91 


33 




46 


8 




1 


-17 




180 


82 


66 


135 


57 


46 


90 


32 


26 


45 


7 


6 





-18 


-14 


179 






134 






89 




i 


44 






- 1 






178 


81 


65 


133 


56 


45 


88 


31 


25 


43 


6 


5 


- 2 


-19 


-15 


177 






132 






87 






42 






- 3 






176 


80 


64 


131 


55 


44 


86 


30 


24 


41 


5 


4 


- 4 


-20 


-16 


175 






130 






85 






40 






- 5 






174 

173 


79 


63 


129 

128 


54 


43 


84 
83 


29 


23 


39 
38 


4 


8 


- 6 

- 7 


-21 


-17 


172 


78 


62 


127 


53 




82 


28 




37 


3 


2 
1 


- 8 


-22 




171 
170 


77 


126 
125 


52 


42 


81 
80 


27 


22 


36 

35 


2 


- 9 
-10 


-23 


-18 


169 


76 


61 


124 


51 


41 


79 


26 


21 


34 


1 


-11 


-24 


-19 


168 






123 






78 




1 


83 






-12 
-13 


-25 


-20 



20* 



234 



"Weight of Substances of Construction. 



Table of the Weight of Substances of Construction, shovnng the 
loeight of a cubic inch, and a cubic foot, in ounces and pounds avoir- 
dupois, and also the number of cubic inches in one pound, of the 
substances most losed in construction. 





Weight of 


"a cubic foot. 


1 Weight of a 


cubic inch. 


■ 


TMamp« nTRfirliP^ 










^iimlipr nf 


AlUlXt\,>S \Jl l_#\J\AlWO* 










cubic inches 




in oz. 


in lbs. 


inoz. 


in lbs. 


in a pound. 


Copper, cast, . 


1 8788 


549-25 


5-086 


•3178 


3-146 


Copper, sheet, . 


8915 


557-18 


5-159 


•3225 


3-103 


Brass, cast, . . 


' 8396 


524-75 


4-852 


•3037 


3-293 


Iron, cast, . . 


; 7271 


445-43 


4-203 


•263 


3-802 . 


Iron, bar, . . 


' 7631 


476-93 


4-410 


•276 


3-623 . 


Lead, .... 


11344 


709-00 


6-456 


•4103 


2-437 


Steel, soft, . . 


7833 


489-56 


4-527 


•2833 


3-530 


Steel, hard, . . 


7816 


488-50 


4-517 


•2827 


3-537 


Zinc, cast, . . 


7190 


449-37 


4-156 


•26 


3-845 


Tin, cast, . . . 


7292 


455-75 


4-215 


•2636 


3-790 


Bismuth, . . . 


9880 


619-50 


5-710 


•3585 


2-789 


Gun-metal, . . 


8784 


549-00 


5-0075 


•3177 


3-147 


Sand, .... 


1520 


95-00 


-8787 


-055 


18-190 


1 Coal, .... 


1250 


78-12 


-7225 


•0452 


22-120 - 


1 Brick, .... 


2000 


125-00 


1-156 


•0723 


13-824 


i Stone, paving, . 


2416 


151-00 


1-396 


•0873 


11-443 


Slate, .... 


2672 


167-00 


1-544 


•0967 


10-347 


Marble, . . . 


2742 


171-37 


1-585 


•0991 


10-083 


White lead, . . 


3160 


197-50 


1-826 


•1143 


8-750 


Glass, .... 


2880 


180-00 


1-664 


•1042 


9-600 


Tallow, . . . 


945 


59-06 


•5462 


•0087 


29-258 


Cork, .... 


240 


15-00 


-138 


-0197 


115-200 


Larch,. . . . 


544 


34-00 


-315 


-0201 


50-823 ! 


Elm, .... 


556 


34-75 


•321 


•0201 


49-726 ! 


Pine, pitch, . . 


660 


41-25 


-382 


•024 


41-890 1 


Beech, . . . 


696 


43-50 


-403 


•0252 


39-724 


Teak, .... 


745 


46-56 


-431 


•027 


37113 


Ash, .... 


760 


47-50 


-440 


•0275 


36-370 i 


Mahoarany, . . 


1 852 


53-25 


-493 


•0308 


32-449 1 


Oak,^. \ - . 


1 970 


60-62 


•561 


•0351 


28-505 1 


Oil of turpentine 


! 870 


54-37 


•503 


•0315 


31-771 


Olive oil, . . . 


915 


57-18 


•529 


•0331 


30-220 


Linseed oil, . . 


932 


58-25 


•539 


•0337 


29-655 


Spirits, proof, . 


927 


57-93 


-536 


•03352 


29-288 


Water, distilled, 


1000 


60-50 


•578 


•03617 


27-648 


" sea, . . 


1028 


64-25 


•594 


•0372 


26-894 


Tar, .... 


1015 


63-43 


•587 


•0367 


27-242 


Vinegar, . . . 


1026 


64-12 


•593 


■037 


26-949 


Mercury, . . . 


13568 


848-00 


7-851 


•4908 


2-037 



Solders. 



235 



Conducting Power of Materials used in the Constructio7i of Houses. 
As observed by Mr. Hutchinson. 



Slate, .... 
Keene's cement, 
Plaster and sand, 
Plaster of Paris, 
Roman cement, 
Beech wood, 
Lath and plaster, 
Fir wood. 



100 
19-01 
lS-10 
20-26 
20-80 
22-44 
25-55 
27-60 



Oak wood, 33-66 

Asphalt, 45-19 

Chalk (soft), . . . • 56-38 

Stock brick, .... 60-14 

Bathstone, 61-08 

Fire brick, 61-70 

Lead, 521-34 



Air and gases are very imperfect conductors. Heat appears to 
be propagated through them almost entirely by conveyance, the 
heated portions of air becoming lighter, and diffusing the heat 
through the mass in their ascent as in liquids. Hence, in heating a 
room with hot air, the hot air should be introduced at the lowest 
part. The advantage of double windows for Avarmth depends, in a 
great measure, on the sheet of air confined between them through 
which heat is very slowly transmitted. 

Capacity/ of Bodies for Tra7ism.it ting Heat. 

The capacity which bodies possess of transmitting heat, does not 
depend upon their transparency ; or bodies are not all transparent 
to heat in the same proportion that they are transparent to light. 
The following plates of an equal thickness of '1031 inches allowed 
very different proportions of heat to pass through them. 

Of 100 rays transmitted from an Argand oil lamp there were : 



Rock salt, 92 

Mirror glass, 62 

Rock crystal, 62 

Iceland spar, 62 

Rock crystal, smoky & brown 57 

Carbonate of lead, .... 52 

Sulphate of barytes, ... 33 



Emerald, 29 

Gypsum, 20 

Fluor spar, 15 

Citric acid, ...... 15 

Rochelle salt, 12 

Alum, .12 

Sulphate of copper, ... 



SOLDEES. 

For Lead. — Melt one part of block tin, and, when in a stote of 
fusion, add 2 parts of lead. Resin should be used with this solder. 

For Tin. — Pewter, 4 parts ; tin, 1 ; bismuth, 1. Melt them 
together and run them into slips. Resin is also used with this 
solder. 

For Gold. — Pure gold, 12 parts; silver, 2; copper, 4. 

For Brass. — Brass, 2 parts; zinc, 1. 

For Iron. — Good tough brass, with a small quantity of borax. 

For Pewter. — Bismuth, 2 parts; lead, 1 ; tin, 2. 

For Copper. — Copper, 2 parts ; zinc, 1. 

For Silver. — ^Silver, 5 parts ; brass, 6 ; zinc, 2. 

Hard Solder. — Cop[)cr, 2 parts ; zinc, 1. 

Soft Solder. — Tin, 2 parts; lead, 1 part. 



236 


Gradations of 


Temperature. 






TABLE 


Of proportions for making Shafti?)g vnth Half -lap Couplings, shomng 


length Oj 


f Xeck and sizes of Coupling-box. {Manchester Rides.) 


Diameter of 


Length of 


Diameter of 


! 

Length of Length of \ Diameter of 


Neck. 


Neck. 


Coupling. 


Lap. 


Box. Box. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. Inches. 


2 


4 


3i 


2i 


o\ h\ 


2i 


^ 


3J 


2f 


6 ' 6 


2i 


5 


4 


8 


6i ! 6f 


2f 


H 


^ 


3i 


V 


n 


3 


6 


4i 


3^ 


H 


7f 


3i 


H 


5 


3| 


8 


8i 


H 


H 


— 




— 




4 


>1 


6 


4 


8i 


n 


4i 


n 


6i 


4i 


9 


lOi 


6 


8 


H 


5 


91 ■ Hi 


H 


8i 


8i 


o\ 


11 


12i 


6 


9 


9 


6 


12 


13^ 


6i 


9J 


9f 


6i 


13 


14f 


7 


lOir 


lOi 


Vi 


14 


16 


n 


Hi 


— 







-- 


8 


12 


12 


8 


16i 


18 


8* 


12* 


12i 


8i 


17 


19 


9 


13i 


13 


9 


18 


20 


9^ 


14 


— 








^- 


10 


14i 


14 


10 


18^ 1 


22 


11 


15 


16 


11 20 1 


24 


12 


16 


in . 


12 


21 1 


26 



Gradations of Temperature. 
The following are interesting facts in the range of temperature : 
166° Greatest artificial cold. (Faraday.) 
150 Liquid nitrous oxide freezes. 
122 Liquid sulphuretted hydrogen freezes. 
105 Liquid sulphurous acid freezes. 
91 Greatest artificial cold measured by Walker. 
56 Greatest natural cold observed by a " verified " ther- 
mometer. (Sabine.) 
70 Greatest natural cold observed at Fort Reliance by Back. 

(Doubtful.) 
58 Estimated temperature of planetary space. (Fourier.) 
47 Sulphuric ether freezes. 
^ 39 Mercury freezes. 

30 Liquid cyanogen freezes. (Faraday.) 

13 Mean temperature at the Pole. (Arago.) 

11 A mixture of two parts alcohol and one part water freezes. 

7 A mixture of equal parts alcohol and water freezes. 



r 



Gradations of Temperature. 237 



/ 20° 


1 
Strong wine freezes. 




28 


Vinegar freezes. 




80 


Milk freezes. 




32 


Ice melts. 




41 


Mean temperature at Edinburgh. 




50-7 


Mean temperature of London. 




60 


Mean temperature at Rome. 




81-5 


Mean temperature at the equator. 




- 98 


Heat of the human blood. 




98 


Ether boils. 




100 


Phosphorus melts. 




173 


Alcohol boils. 




117 


Highest natural temperature observed of a hot wind 
in Upper Egypt. (Burckhardt) 




133 


Wood-spirit boils. 




142 


Spermaceti melts. 




151*34 Beeswax melts. ^ | 




212 


Water boils. 




226 


Sulphur melts. 


■^ 


242 


Nitric acid boils. 


1 


283 


A compound of equal parts of tin and bismuth melts. 




442 


Tin melts. 


T. 


460 


The surface of polished steel acquires a pale straw 


N ^ 




color. 




476 


Bismuth melts. 


< 


554 


Phosphorus boils. 


560 


Oil of turpentine boils. 




580 


The surface of polished steel acquires a uniform deep 
blue. 




590 


Sulphuric acid boils. (Dal ton.) 




594 


Lead melts. 




600 


Linseed oil boils. 




635 


Lowest ignition of iron in the dark. 




662 


Mercury boils. 




700 


Zinc melts. 




752 


Iron bright red in the dark. 




810 


Antimony melts. 




884 


Iron red hot in the twilight. 




1077 


Red heat fully visible in the daylight. 




1141 


Heat of a common fire. (Daniell.) 




1869 


Brass melts. 




1873 


Silver melts. 




1996 


Copper melts. 




2016 


Gold melts. 




2500 


Steel melts. 




2786 


Cast-iron melts. 




\3080 


Platinum melts. 


1 


Mie line 


' of perpetual congelation has a variable altitude in 


diff 


erent c 


imatcs. j 



238 Properties of Lumbers. 



At the equator it is 14760 feet. 

At the AJps " 8120 " 

In Iceland " 3084 " 

At the polar regions ice is perpetually observed at the surface of 
the earth. 



PEOPEETIES OF NUMBEES. 

1. A Prime Number is that which can only be measured by 1 or 
unity. 

2. A Composite Number is that which can be measured (or divided 
without a remainder) by some number greater than unity. 

3. A Perfect Number is that which is equal to the sum of all its 
divisors, or aliquot parts : thus 6 = | + | + f . 

4. If an odd number divides an even number, it will also div^ide 
the half of it. 

5. If the last digit of any number be divisible by 2, the whole 
number is divisible by 2. 

6. If the two last digits be divisible by 4, the whole number is 
divisible by 4. 

*1. If the three last digits be divisible by 8, the whole number is 
divisible by 8. 

8. If a number terminate with 5, it is divisible by 5 ; and if it 
terminate with 0, it is divisible by either 10 or 5. 

9. If the sum of the digits constituting any number be divisible 
by 3 or 9, the whole is divisible by 3 or 9 ; and if also the last digit 
is even, the whole number is divisible by 18. 

10. If the sum of the digits of any number be divisible by 6, and 
the right hand digit by 2, the whole is divisible by 6. 

11. If the sum of the 1st, 3d, 5th, <fee., digits of any number be 
equal to that of the 2d, 4th, 6th, &c., that number is divisible by 11. 

Thus 327943 contains 11 = 29813 times exactly. 

12. If a square number be either multiplied or divided by a 
square, the product or quotient is a square ; and conversely, if a 
square number be either multiplied or divided by a number that is 
not a square, the product or quotient is not a square. 

13. The product arising from two different prime numbers cannot 
be a square number. 

14. The product of no two different numbers prime to each other 
(that is, 1 being the common measure) can make a square, unless 
each of those numbers be a square. 

15. The square root of an integral number, that is not a complete 
square, can neither be expressed by an integer nor by any rational 
fraction ; so with the cube root of an integer. 

16. Every prime number greater than two, is made up of 4 times 
some number, + 1 or — - 1 ; that is, of one of the forms 4cn + 1, or 

4:71 — 1. 











Properties of 


Numbers 




239 


IT. Any prime number greater than 3, divided by 6, will leave a 
remainder of 1 or 5 : that is, every number greater than 3, is one 
of the forms 67i + ], or 6/^ — 1. 

18. The number of prime numbers is infinite. 

19. A square number cannot terminate with an odd number of 
cyphers. 

20. If a square number terminate with 4, the last figure but one 
will be an even number. 

21. If a square number terminate with 6, it will terminate with 25. 

22. Ko square number can terminate with two equal digits, 
except two cyphers, or two fours. 

23. Iso number whose last digit is 2, 8, '7, or 8, is a square number. 
^ 24. If a cube number be divisible by 7, it is also divisible by the 
cube of 7. 

25. The difference between any integral cube and its root is 
always divisible by 6. 

26. Neither the sum nor the difference of two cubes can be a 
cube. 

27. A cube number may end with any of the natural numbers. 

28. All the powers of any number that end with 6, w411 terminate 
with 6 ; so with the numeral 5. 

TABLE ' 

Of the first Nine Powers of the first Nine Numbers. 


1st 


2d 


3d 


4th 


5th 6th 


7th 


8th 


1 
9th 


1 


1 


1 


1 


1 1 


1 


1 


1 


2 


4 


8 


16 


32 64 


128 


256 1 512 


3 

7 


9 
16 


27 


81 


243 729 


2187 


6561 


■ 19683 


64 


256 


1024 4096 


16384 


65536 


262144 


5 


25 


125 


625 


3125 15625 


78125 


390625 


1953125 


6 


86 


216 1296 


7776 


46656 


279986 


1679616 


10077696 


7 
8 
9 


49 
64 

81 


348 


2401 


16807 


117649 


823548 


5764801 


40353607 


512 


4096 


32768 


262144 


2097152 


16777216 


134217728 


729 


6561 


59049 


581441 


4782969 


43046721 


887420489 









240 



Useful Numbers. 



Log. 
Log. 



1 



1 



TABLE 

Of Useful Numbers. 
=r. 8'14] 592Y -v/7' . 



. 0-49'7l499 
. 1*1447299 

. 0-3183099 
. 9-8696044 
. 0-1013212 
. l-'7'724538 
.0-5641896 



1 






= 1^4142136 
. 0-7071068 



4-4428829 
2-2214415 



"^ ^ 0-4501582 






.1-2533141 

.'a-7978846 



^og. e 

Modulus of common logarithms 

Log. of ditto 

9 



/r- 

Log. g o .. . 

Inches in a French metre 

Log. of ditto 

Feet in ditto . 

Log. of ditto 

Square feet in the square mi^tre 

Acres in the Are 

Lbs. in a kilogramme 

Log. of ditto 

Imperial gallons in a litre 

Lbs. per square inch in 1 kilogramme per square 

millimetre 

Cwts. ditto, ditto 

Volume of a sphere whose diameter is 1 . . . . 

Arc of r to rad. 1 

Arc of r to rad. 1 

Arc of 1" to rad. 1 

Degrees in an arc whose length is 1 

Grains in 1 oz. avoirdupois 



= 2-7182818 
0-4342945 

-434294482 
9-6377843 

32-19084 
5-67363 
1-5077222 

89-37079 
1-5951741 
3-2808992 
0-5159929 

10 764297 
0024711 
2-20548 
0-3435031 
02200967 

1422 
12-7 

0-5235988 
0017453293 
0000290888 
0-000004848 
57-295780° 
437i 



Surface of Boilers' Tubes. 



241 



Grains in 1 lb. ditto 7000 

Grains in a cubic inch of distilled water, Bar. 30 

in., Th. 62° 252-4o8 

Cubic inches in an ounce of water 1 •'73298 

Cubic inches in the imperial gallon 277*276 

Feet in a geographical mile 6075*6 

Log. of ditto 3*7835892 

Feet in a statute mile 5280 

Log. of ditto 3*7226339 

Length of seconds' pendulum in inches .... 39*19084 

Cubic inches in 1 cwt. of cast iron 430*25 

Bar iron , 397*60 

" " Cast brass 368*88 

" " Cast copper 352*41 

" " Cast lead ...... 272*80 

Cubic feet in 1 ton of paving stone . . - . * . 14*835 

** ** Granite : . 13*505 

*' " Marble 13*070 

« " Chalk 12*874 

" " Limestone 11*273 

*■' ** Elm 64*460 

" " Honduras mahogany . . . 64*000 

" *' Mar Forest fir 51*650 

" " Beech 51*494 

** " Kigafir 47*762 

" " Ash and Bantzic oak . . . 47*158 

" " Spanish mahogany .... 42*066 

« " English oak 36*205 

To find th^ weight in lbs. of 1 foot of common 
rope, multiply the square of its circumference 

in inches by -044 

to -046 

Ditto for a cable '027 

TABLE 
Surface of Boilers' T'uhes of Different Lengths and Diameters. 



Diameter. 


Length. 


Surface. 


Diameter. 


Length. 


Surface. 


In. 


Ft. in. 


Sq. ft. 


In. 


Ft. in. 


Sq. ft. 


2i 


5 


3*27 


3 


6 6 


51 


u 


5 3 


8-42 




6 8 


52 


u 


5 6 


3*6 




7 


5-5 


u 


5 9 


3*75 




7 6 


5-89 


" 


6 


3-9 




8 


6-28 


3 


6 


4*7 




8 6 


6-67 


it 


6 3 


4-9 









21 



242 Recipes for Making Glass. 



EECIPES FOE MAKING DIFFERENT j 
KINDS OF GLASS. | 

1. Bottle Glass. — 1. Dry glaiiber salts, 11 pounds; soa per salts, 12' 
pounds; half a bushel of waste soap ashes; sand, 5G pounds; glass I 
skimmings, 22 pounds ; green broken glass, 1 cwt. ; basalt, 25 
pounds. This mixture affords a dark green glass. 

2. Yellow or white sand, 100 parts ; kelp, 30 to 40 ; lixiviated 
wood ashes, from 160 to 170 parts; fresh wood ashes, 30 to 40 
parts; potter's clay, 80 to 100 parts; cullet, or broken glass, 100. 
If basalt be used, the proportion of kelp may be diminished. 

2. Green Window, or Broad Glass. — Dry glauber salts, 11 pounds ; 
soaper salts, 10 pounds ; half a bushel of lixiviated soap waste ; 
50 pounds of sand ; 22 pounds of glass pot skimmings ; 1 cwt. of 
broken green glass. 

3. Crown Glass. — 300 parts of fine sand ; 200 of good soda ash ; 
33 of lime ; from 250 to 300 of broken glass; 60 of white sand; 30 
of purified potash; 15 of saltpetre; (1 of borax;) ^ of arsenious 
acid. 

4. Nearly White Table Glass.— 1. 20 pounds of potashes; 11 pounds' 
of dr}' glauber salts; 16 of soaper salt; 55 of sand ; 140 of cullet 
of the same kind. 

2. 100 parts of sand; 235 of kelp; 60 of wood ashes; 1|- of 
manganese; 100 of broken glass. 

5. White Table Glass. — I. 40 pounds of potashes; 11 of chalk ;i 
7o of sand ; -| of manganese ; 95 of white cullet. 

2. 50 of purified potashes ; 100 of sand ; 20 of chalk, and 2 of 
saltpetre. 

6. Crystal Glass. — 1. 60 parts of purified potashes ; 120 of sand; 
24 of chalk ; 2 of saltpetre ; 2 of arsenious acid ; -Jg- of manga- 
nese. 

2. Purified pearlashes, 70 parts ; white sand, 120; saltpetre, 10; 
■J of arsenious acid ; i- of manganese. 

3. 67 of sand : 23 of purified pearlashes ; 10 of sifted slaked 
lime ; J of manganese ; 5 to 8 of red lead. 

4. 120 of white sand; 50 of red lead; 40 of purified pearlash ; 
20 of saltpetre ; ^ of manganese. 

5. 1 20 of white sand ; 40 of pearlash purified ; 35 of red lead ; 
13 of saltpetre; -J^ of manganese. 

6. 30 of the finest sand; 20 of red lead; 8 of pearlash purified; 
2 of saltpetre; a little arsenious acid and manganese. 

7. 100 of sand ; 45 of red lead ; 35 of purified pearlashes ; \ of 
manganese ; I of arsenious acid. 

7. Plate Glass — 1. Very white sand, 300 parts ; dry purified soda, 
100 parts; carbonate of lime, 43 parts ; manganese, 1 ; cullet, 300. 

2. Finest sand, 720 parts: purified soda, 450; quicklime, 80; 
saltpetre, 25 ; cullet, 425. 

A little borax has also been prescribed ; much of it communi- 
cates an exfoliating property to glass. 







Priml: Numbers. 




24« 








TABLE 










Of Prime Nanibers to 5000. 




2 


197 


461 


751 


1051 


1381 


1697 


3 


199 


463 


757 


1061 


1399 


1699 


5 


211 


467 


761 


1063 


1409 


1709 


7 


223 


479 


769 


1069 


1423 


1721 


11 


227, 


487 


773 


1087 


1427 


1723 


13 


229 


491 


787 


1091 


1429 


1733 


17 


233 


499 


797 


1093 


1433 


1741 


19 


239 


503 


809 


1097 


1439 


1747 


23 


241 


509 


811 


1103 


1447 


1753 


29 


251 


521 


821 


1109 


1451 


1759 


31 


257 


523 


823 


1117 


1453 


1777 


37 


263 


541 


827 


1123 


1459 


1783 


41 


269 


547 


829 


1129 


1471 


1787 


43 


271 


557 


839 


1151 


1481 


1789 


47 


277 


563 


853 


1153 


1483 


1801 


53 


281 


569 


857 


1163 


1487 


1811 


59 


283 


571 


859 


1171 


1489 


1823 


61 


293 


577 


863 


1181 


1493 


1831 


67 


307 


587 


877 


1187 


1499 


1847 


71 


311 


593 


881 


1193 


1511 


1861 


73 


313 


599 


883 


1201 


1523 


1867 


79 


317 


601 


887 


1213 


1531 


1871 


83 


331 


607 


907 


1217 


1543 


1873 


89 


337 


613 


911 


1223 


1549 


1877 


97 


347 


617 


919 


1229 


1553 


1879 


101 


349 


619 


929 


1231 


1559 


1889 


103 


353 


631 


937 


1237 


1567 


1901 


107 


359 


641 


941 


1249 


1571 


1907 


109 


367 


643 


947 


1259 


1579 


1913 


113 


373 


647 


953 


1277 


1583 


1931 


127 


379 


653 


967 




1279 


1597 


1933 


131 


383 


659 


971 




1283 


1601 


1949 


137 


389 


661 


977 




1289 


1607 


1951 


139 


397 


673 


983 




1291 


1609 


1973 


149 


401 


677 


991 




1297 


1613 


1979 


151 


409 


683 


997 




1301 


1619 


1987 


157 


419 


691 


1009 




1303 


1621 


1993 


163 


421 


701 


1013 




1307 


1627 


1997 


167 


431 


709 


1019 




1319 


1637 


1999 


173 


433 


719 


1021 


1321 


1657 


2003 


179 


439 


727 


1031 


1327 


1663 


2011 


181 


443 


733 


1033 


1361 


1667 


2017 


191 


449 


739 


1039 


1367 


1669 


2027 


193 


457 


743 


1049 


1373 


1693 


2029 



244 




Prime Ts'umbers. 






2039 


2399 


2789 


3203 


3581 


3967 


4371 


2053 


2411 


2791 


3209 


3583 


3989 


4391 


2063 


2417 


2797 


3217 


8593 


4001 


4397 


2069 


2423 


2801 


3221 


3607 


4003 


4409 


2081 


2437 


2803 


3229 


3613 


4007 


4421 


2083 


2441 


2819 


3251 


8617 


4013 


4423 


2087 


2447 


2833 


3253 


3623 


4019 


4441 


2089 


2459 


2837 


3257 


3631 


4021 


4447 


2099 


2467 


2843 


3259 


3637 


4027 


4451 


2111 


2473 


2851 


3271 


3643 


4049 


4457 


2113 


2477 


2857 


3299 


3659 


4051 


4463 


2129 


2503 


2861 


3301 


3671 


4057 


4481 


2131 


2521 


2879 


3307 


3673 


4073 


4483 


2137 


2531 


2887 


3313 


3677 


4079 


4493 


2141 


2539 


2897 


3319 


3691 


4091 


4507 


2143 


2543 


2903 


3323 


3697 


4093 


4513 


2153 


2549 


2909 


3329 


3701 


4099 


4517 


2161 


2551 


2917 


3331 


3709 


4111 


4519 


2179 


2557 


2927 


3343 


3719 


4127 


4523 


2203 


2579 


2939 


3347 


8727 


4129 


4547 


2207 


2591 


2953 


3359 


3733 


4133 


4549 


2213 


2593 


2957 


3861 


3739 


4139 


4561 


2221 


2609 


2963 


3371 


3761 


4153 


4567 


2237 


2617 


2969 


3373 


3767 


4157 


4583 


2239 


2621 


2971 


3389 


3769 


4159 


4591 


2243 


2633 


2999 


3391 


3779 


4177 


4597 


2251 


2647 


3001 


3407 


3793 


4201 


4603 


2267 


2657 


3011 


3413 


3797 


4211 


4621 


2269 


2659 


3019 


3433 


3803 


4217 


4637 


2273 


2663 


3023 


3449 


3821 


4219 


4639 


2281 


2671 


3037 


3457 


3823 


4229 


4643 


2287 


2677 


3041 


3461 


3833 


4231 


4649 


2293 


2683 


3049 


3463 


3847 


4241 


4651 


2297 


2687 


3061 


3467 


8851 


4248 


4657 


2309 


2689 


3067 


3469 


3853 


4253 


4663 


2311 


2693 


3079 


3491 


3863 


4259 


4673 


2333 


2699 


3083 


3499 


3877 


4261 


4679 


2339 


2707 


3089 


3511 


3881 


4271 


4691 


2341 


2711 


3109 


3517 


3889 


4273 


4703 


2347 


2713 


3119 


3527 


3907 


4283 


4721 


2351 


2719 


3121 


3529 


3911 


4289 


4723 


2357 


2729 


3137 


3533 


3917 


4297 


4729 


2371 


2731 


3163 


3539 


3919 


4327 


4733 


2377 


2741 


3167 


3541 


3923 


4387 


4751 


2381 


2749 


3169 


3547 


3929 


4339 


4759 


2383 


2753 


3181 


3557 


3931 


4349 


4783 


2889 


2767 


3187 


3559 


3943 


4357 


4787 


2393 


2777 


3191 


3571 


3947 


4363 


4789 



Solid Inches and Solid Feet. 



245 



4793 
4799 
4801 
4813 



4817 
4831 
4861 

4871 



4877 
4889 
4903 
4909 



4919 
4931 
4933 

4937 



4943 
4951 
4957 

4967 



4969 
4973 

4987 
4993 



4999 
5003 
5009 



TABLE 

Of Solid Liches and Solid Feet. 





Inches. 


Feet. 






Inches. 


Feet. 




Feet. 


Inches. 


Feet. 


Inches. 


1 = 


r: 1728 


26 = 


=44928 


51r 


- 88128 


76r 


r]31328 


2 


3456 


27 


46656 


52 


88956 


77 


133056 


3 


5184 


28 


48884 


58 


91584 


78 


134784 


4 


6912 


29 


50112 


54 


93312 


. 79 


136512 


5 


8640 


30 


51840 


55 


95040 


80 


138240 


6 


10368 


31 


53568 


56 


96768 


81 


139968 


7 


12096 


32 


55296 


57 


98496 


82 


14lor6 


8 


13824 


33 


57024 


58 


100224 


83 


143424 


9 


15552 


34 


58752 


59 


101952 


84 


145152 


10 


17280 


35 


60480 


60 


103680 


85 


146880 


11 


19008 


36 


62208 


61 


105408 


86 


148608 


12 


20736 


37 


63936 


62 


107136 


87 


150336 


13 


22464 
24192 


38 
39 


65664 
67392 


63 


108864 


88 


152064 


14 


64 


110592 


89 


153792 


15 


25920 


40 


69120 


65 


112320 


90 


155520 


16 


27648 


41 


70848 


66 


114048 


91 


157248 


17 


29376 


42 


72576 


67 


115776 


92 


158976 


18 


81104 


43 


74e304 


68 


117504 


93 


160704 


19 


82832 


44 


76032 


69 


119232 


94 


162432 


20 


34560 


45 


77760 


70 


120960 


95 


164160 


21 


36288 


46 


79488 


71 


1226b8 


96 


165888 


22 


38016 


47 


81216 


72 


124416 


97 


167616 


23 


39744 


48 


82944 


73 


126144 


98 


169344 


24 


41472 


49 


84672 


74 


127872 


99 


171072 


25 


43200 


50 


86400 


75 


129600 


100 


172800 



TABLE 

Shoiving the Weight of Cast-iron Platen, 12 inches square, and from 

^ of an inch to 1 inch thick. 



One 
Inch. 



Width in 
Inches. 


•125 


i 
•25 


1 

375 


•5 


•625 


3 

4 

•75 


1 
•875 


12 


lbs. oz 
4 13f 


lbs. oz. 
9 lOi- 


lbs. oz. 
14 8 


lbs. 02 

19 5f 


lbs. oz 
24 2^ 


lbs. oz 
29 


lbs. oz. 
88 13f 



lbs. oz. 
88 10} 



21* 



246 



Dimensions of Wheels. 



To find the Hor&e Power that a Cant-Iron Wheel is capable of 
iransmittinff. 

Multiply the breadth of the teeth or face of the wheel in inches 
by the square of the thickness of one tooth, and divide by the 
length of the teeth, for the strength at a velocity of 186 feet per 
minute. 

Thus a wheel with the breadth of teeth = 7^ inches, thickness= 
1*4, and length = 2, ought to transmit 7 '35 horse power. For 

1-4= = 1-96, and ^'^ ^ ^'^^ = 7 -35. 

The strength at any other velocity is found by multiplying the 
power so obtained by any other required velocity, and by -0044. 

Thus, the wheel as above, at the velocity of 320 feet per minute, 
would be capable of transmitting 10'3488 horse power. 

TABLE 

Of the Dimensionfi of Wheels in Actual Use. 



Pitch in 
inches. 


Character of Wheel. 


Number 

of 

teeth. 


Bre'dth 

in 
inches. 


No. of 
revolu- 
tions per 


Horse Power. 














minute. 


Actual. 


Calculated. 


H 


Spur Wheel, . . 


72 


^ 


120 


8 


7-5 


2i 


Spur Wheel, . . 


95 


6 


25 


H 


1-676 


3i 


Bevil Wheel, . . 


40 


7 


SOi 


20 


24-34 


21- 


Cog Wheel, . . . 


60 


6 


40 


12 


15-82 


H 


Bevil Wheel, . . 


• 70 


12 


10 


60 


67-396 


n 


Spur Wheel, . . 


90 


8 


12 


6 


9-72 


3f 


Internal, .... 


80 


9 


20 


41 


48-8 


3 


Cog Spur Wheel, . 


60 


8 


30 


121 


•177 


6 


Spur Wheel, . . 


30 


14 


7 


21 


-26i 


4 


Spur Wheel, . . 


100 


10 


8 


25 


29-6 


21 


Spur Wheel, . . 


33 


7 


55 


23 


•25 


2f 


Spur Wheel, . , 


108 


7 


20 


25 


•26 


. n 


Internal, . . . 


100 


7 


10 


87 


90-4 


5 


Internal, .... 


60 


12 


12 


55 


53 -5 


5 


Spur, 


41 


10 


20 


61 


•50 


476 


Spur, 


50 


12 


23 


65 


71-3 


3f 


Bevil Wheel, . . 


35 


10 


24 


26 


25-6 


4 


Cog Bevil Wheel, . 


50 


10 


28 


33 


32-6 


4 


Cog Spur Wheel, . 


35 


9 


20 


18 


16-3 


6 


On Water Wheel, . 


112 


14 


12 


110 


•168 


4i 


Spur Wheel, . . 


55 


10 


16 


56 


54-56 





Transverse Strength of Bodies. 



241. 



TABLE 

Showing the CWcumference of a Rope equal to a Chain made of Iron 

of a given Diameter, and the Weight in Tons that each is proved to i 

carry ; also the weight of a Foot of Chain made from Iron of that | 

dimension. \ 



Rope's circum- 
ference in inches. 


Chain Diameter in 
inches. 


Proved to carry 
in tons. 


Weight of a linear 
foot in lbs. avoir. 




i & fV 






3 


1 


1-08 


4 


1 


2 


1-5 


41 


i&i\ 


3 


2 


H 


i 


4 


2-1 


6 


i&T\ 





3-3 


H 


f 


6 


4 


1 


i&3V 


8 


4-6 


H 


i 


n 


0-5 


8 


I&tV 


ni 


6-1 


9 


1 


13 


1-2 


n 


*&tV 


15 


8*4 


lOi 


1 inch. 


18 


9-4 



The Transverse Strength of a body is that power which it exerts 
in opposing any force acting in a perpendicular direction to its | 
length, as in the case of beams, levers, <fec., it is inversely as their 
lengths, and directly as their breadths, and the square of their j 
depths. But, if cylindrical, as the cubes of their diameters. i 

That is, if a beam 5 feet long, 2 inches broad, and 3 inches deep, ! 
can carry 1798 lbs., another beam of the same material, 10 feet 
long, 2 inches broad, and 3 inches deep, will only carry 899 lbs., 
being inversely as their lengths. 

Again, if a beam 5 feet long, 2 inches broad, and 3 inches deep, 
can support 1798 lbs., another beam of the same material, 4 inches 
broad, and 3 inches deep, will support double that weight, being 
directly as their breadths. | 

A beam of the same material, 5 feet long, 2 inches broad, and 6 
inches deep, will sustain 7192 lbs., being as the square of their 
depths. 



248 


Equivalents and 


Specific Gravities. 












TABLE 






Shoioing the 


Equivalents 


and Sp 


ecijic Gravities 


of sixty-two Simple | 






Substances. 












>. 


\ 






^ 


Name of 


t 




o 


Name of 


2 




i 


Substance. 


a 


>H 


1 , 


Substance. 


s 

>> 


1.2 


^ 




cc 


is 






CQ 


^ g 


o 






H 


Q. 






H 




Hydrogen, . 


H. 


1 


•06^ 


59 Metals 








Oxygen, . . 


Oor. 


8 


l-02( 


5 Continued. 








Nitrogen, 


N. 


14-2 


l-52t 










Chloi'ine, 


CI, 


35-5 


2-44^ 


\: Chromium, 


Cr. 


28-19 


5 


9 


Carbon, . . 


c. 


6-12 


44-1 


Mercury, . 


Hg. 


203 


13 


5 


Iodine, . . 


I. 


126-5 


4-94^ 


J Silver, . . 


^g- 


108-3 


10 


5 


Sulphur, . . 


s. 


16-1 


1-99 


Gold, . . 


Au. 


200 


19 


3 


Phosphorus, . 


p. 


ih-n 


1-7 


Platinum, . 


Pt. 


98-8421 


5 


Fluorine, . . 


R 


18-7 




Tin, . . . 


Sn. 


68-9 


7 


29 


Bromine, 


Br. 


18-4 


3- 


Cobalt, . . 


Co. 


29-5 


7 


83 


Boron, . . 


B. 


11 




Manganese, 


Mn. 


27-7 


8 





Selenium, 


Se. 


40 


4-5 


Nickel, . . 


]Ni. 


29-5 


8 


8 










Antimony, 


Sb. 


64-6 


6 


7 










Arsenic, . 


As. 


37-7 


5 


7 


Metals. 








Palladium, 


Pd. 


53-35 


11-5 










Rhodium, . 


R. 


52-2 


11 


Potassium, . 


K. 


89-2 


•86^ 


) Asmium, . 


Os. 


99-7 


10 


Sodium, . . 


Xa. 


23-5 


-971 


\ Iridium, . 


Ir. 


99-8 


18-68 


Lithium, . . 
Calcium, . . 


L. 

Ca. 


10 

20-5 




Cadmium, . 
Molybde- 


Cd. 


55-8 


8 


6 






Magnesium, . 


Mg. 


12-7 




num, . . 


Mo. 


47-9 


8-6 


Silicon, . . 


Si. 


22 




Tungsten, or 








Aluminum, . 


Al. 


lB-7 




"Wolfram, 


W. 


94-8 


17 


Iron, . . . 


Fe. 


28 


7-7 


Vanadium, 


Y. 


68-5 




Lead, . . . 


Pb. 


103-7 


11-35 


Uranium, . 


U. 


217-2 




Copper, . . 


Cu 


31-7 


8-8 


Titanium, . 


Ti. 


24-5 




Columbium, . 


Cm. 


184-8 




Cerium, 


Ce. 


46 




Glucinum, . 


G. 


26 




Niobium, . 


Kr. 






Yltrium, . . 


Y. 


32 




Pelopium, . 


Pe. 






Zirconium, . 


Zr. 


34 




Norium, 


No. 






Thorinum, . 


Th. 


60 




Didymium, 


D. 






Strontium, . 


Sr. 


43-8 




Lantanum, 


Ln. 


48 




Barium, . . 


Ba. 


68-6 




Jerbium, . 


Tb. 






Bismuth, 


Bi. 


71-5 




Erbium, 


E. 






Tellurium, . 


Te. 


64-2 




Ptutnheium, 


Ru. 


52 




Zinc, . . . 


Z. 


32-3 


Fronr 
6-8 t 
7-1 


I 








i 

i 



Pendulums. 



240 



Th^ Feeding Properties of different Vegetables. 
In comparison with 10 lbs. of hay. 



Hay, . . 
Clover hay, . 
Yetch hay, . 
Wheat straw, 
Barley straw, 
Oat straw, . 
Pea straw, . 
Potatoes, 
Old potatoes, 
Turnips, 



10 

8 

4 

52 

52 

55 

6 

28 

40 

60 



Carrots, . 

Cabbage, 

Pease and beans, 

Wheat, . 

Barley, . 

Oats, 

Rye, . . 

Indian corn, . 

Bran, 

Oil-cake, 



35 

30 to 40 

2 to 8 

5 

6 

5 

5 

6 

5 

2 



Thus 2 lbs. of oil-cake is worth as much as 55 lbs. of oat straw. 



PENDULUMS. 

A pendulum that vibrates seconds, or 60 in the latitude of 

London, is 39-1393 inches long; and V39^139Fx 60=375*36, which 

serves as a constant number for other pendulums; thus, 375*36 

divided by the square root of the pendulum's length, gives the 

number of vibrations per minute ; and divided by the vibrations 

per minute, gives the square root of the length of pendulums. | 

Example 1. — Required the number of vibrations a pendulum of 

25 inches long will make per minute. 

375-36 ^ ^ ., . 

r — == 75*072 vibrations per minute. 

|/25 ^ 

Example 2. — Required the length of a pendulum to make 80 
vibrations per minute. 

375-36 , . , , 

— — — = 4-692^= 22-014864 inches long. 

80 

Table containing the Length of Pendulums to vibrate Seconds in 
various parts of the World. 





Inches. 




Inches. 


At Sierra Leone, . 


. 89-01954 


At New York, . 


. 89-10153 


" Trinidad, . . 


. 8901879 


" Bordeaux, . . 


. 3911282 


" Madras, . . . 


. 39-02630 


" Paris, . . . 


. 39 12843 


" Jamaica, . . 


. 89-03508 


" Edinburgh, . 


. 3915540 


" Rio Janeiro, . 


. 39-01206 


" Greenland, 


. 39-20328 



A pendulum vibrating half seconds in the latitude of London is 
9*8 inches in length ; and for quarter seconds, 2-5 inches. 



250 



Binary Compounds. 



so 
I— t .^ 



^5^ 
^ 



r 



CQ 



<v 



2; " 



X a ^ 

O O) 



c3 rt 

<^ •'S t" 



S « 

V « « 



03 




5 

c3 


t> 




^ 








O 




P^ 






fl 
^ 



: 53- 



O c3 C O 









' > o rH 



M 



■^ o 



c3 
f:3 5 






"■ ^ t; fl 

00 J:; C3 



F o 

si 



?-i o 3 

^ fl "^ 

f^ G c3 






W 






Pr^ 2 O ^ 



o <v ■ 



^ G 



•nJ o 



3 '^^ s: ^Jl 

<-: -*-3 .2 i-G <*-! 
c3 •« ^— o 



C3 p t^ ,^ C3 



i-O <y "l^ ^ Cfi iiH ^^ 



P-i o 



p2r2 



fi S «8 

G <u su 
03 22 o 



^ -f^ w O ^ O C3 Q^.22 <^ ^^ ^ <^^ ;> <D 

^ 5^ I g «3 c3 « g^;g « ^ ^ 't: « ^ n: 

r^ ^**' fl^ I-* r/S »— ' ^ f^^ ^^ ^ . -^ 



^r^ 5.52 >,«> SS^ 

2 i 



2 ^ « o 2 



O rr-. i=^ ^ 






<v 















= 5 > 



C<1 3<J 

Oi J:- (>1 O 



cq ^ CM d 



o^^ob booowo boo Ww^ 



• <i> <x» 5t 
^5 ^ is 



^ O 

53 , 



I Orrf 



P^/ 



HD 



I ^ O '>< 



-*^ 'c^ 'T^ 

•f r " 






0.2 O 

tc is a 



^ to '^ w z: 

J::^ c3 t; s: 2 a, 



• a 
o 

9 2^ 



53 ,. . «^ 



WWPQ 



Binary Compounds. 



261 



o 

O 






g » > 

=^ 5^ 

..^ 52 <l^ 

^ <^ 

^ £"" 

• > S o 

<aj d r' . 

^ g o ^ 



c o 

c3 O 
g.J 

re P 



O 

a. 



^.2 

w1 






c3 S 

t— ( a> 

> 

03 O 






. ^ C3 p 
a; p ;^ ^ 



^ i I ti) 



C C rH 



^ C3 









02 



CC 



q; OQ O) 
TO f< '^ 

WW P 



^ 2 C^ O 

§ S.g.^S.2 
P Q O 






"■ tc I* ai 

3 O" f> f-i 

S ^^ 

O) C) to 



O <1> 
CO 

.'^^ Oh 
^^ 

^. ^ 

-^ E ® S 

^ '^ -^ n3 o) 

§ 2.2 ^'^ 

<y p_j 1:5 *— i (X) 

,r c3 a G o 

S: "t: O C= <3; 
^ OJ c3 c3 ^ 



u ^: 



- WW fl K >=; 

3 '^ 'o . rf ^ 



&I3 

03 . 



pq 






*5 o 



1*0 o 

I^- CO 






»p ^ O J;- tH T^^ 
CO CO r^ 05 rH lb 

CO OS (M rH J:>- 10 
.-I CO rH 









5J 4) Q? <D 0) (P 



ho 
O 



■^ l-T? 


.s 


n 


<y 


SS'o 


3*^ 


^ * 



1;^ 


-•J S3 









icarbure 
oracic a 
hlorous 




,x3 • 


p 
a 

g 


PQPQO 


oW 


a 


;z4 



b =° 

*o "^ "^ 



H OS'S c3 
2 O 03 otJ 



7? -C2 rC 

_ . ? O O 



hShWWchPh 



n3 



o 
Ph 



T3 



. 
O ^ 

• ^ a - 

• "^-^ 2 

■fl'S d o g 

^ r2 r2 O 2 

q^ Qj <P &.< 1? 

GQ CZ2 :/} Ph P-i 



252 






Binary Compounds. 










^ fl f-> 


cS 








o -S o 










c -2 


'S 








rt 5 o 


*3 










§! ■ 








•S = Ifg g 


.s3 




i 




is formed when iron is oxidated 
:h water at a high temperature, 
d litharge. Used in flint glass. 

Much employed as a pigment. 

e of copper. Native productio 

in crystals of a red color. 

de of copper, or copper black. 

nation of oxj^gen and zinc we kn 
commonly called oxide of Antim 
h alkalies by fusing them togeth< 

id black oxide of tin; great a 


0^ 




I 




;enerally associate 
mosaic gold. Us 
d bronze powder. 


^ 


•5 




•m pound 
ntact wil 
>nly calle 

red lead. 

red oxid 
er mines 
black oxi 

ly combi] 
native ; 
bines wit 

Qies calk 


en. 

native, g 
•ly called 
ze, terme 


< 


«0 




This CO 

in CO 
Commc 

Called 

Called 

copp 

Called 

The on 
Occurs 
It coml 

Sometii 


oxyg 
Occurs 
Former 

bron 


H 


1 

,5> 
















*§ 


^^ 


j^<^^^^ ^^C0G<l(>q(?^ Oi 


CirH 




^^ 


•C-S 


CO T^OOOOs*^ Oir-OCOf^Ci o 


T^ ^ 




>^ 


aM 


r-l r-l.,-,Ttir-lX:- CO ^ ^ \0 iO ro <X> 


i:- C5 




rl rH (M JO ^-1 ,-.■,-( r-H 






•^ 
















« 










i 

4 


1 

1 


"k ££§£5 66^^^¥^ & 


cccc 




6 «o6bo 050^55 


0^ 




1 




1 


• . .-0 f • J • 


• • 




^ 


'O* 


lack oxide of iron, 

rotoxide of lead, . 
inoxide of lead, . 
uadrotisoxide of le 
inoxide of lead, . 
inoxide of copper, 

rotoxide of copper, 
inoxide of copper, 
rotoxide of zinc, . 
3squioxide of antira 
ntimonious acid, . 
ntimonic acid, . . 
rotoxide of tin, or 
num, 


• • 






s 


'3 

0*9 

a « 








pq p,PC?pqp |2,pq(iHgQ<:<^|iH 


S3 



Binary Compounds. 



C3 tJD 






P^ O. 









G .12 

_^ .t: ^ 

■1^ G r2 
^ 2rr-( 

O c8 



a 

CO 

O 






n3 



^ 
fl 






S 
C3 



O 



C3 (D 

15 






Cj O 03 

»^ , r-5 I— I 



C ^ S 

S CO o 
=« ^ ^ 



<v 






^ D 



03 p P 

e ^ o 

C3.S ^t; .,, p 0^':SQ =3 Cp 



o 



-G 



B 5* 



^ >- ^ 



c^ 6 ^ 



^5 '-^ G ^f.^ o -5 i^ -r ^ go:-* 

^ •!-! rr o .rN "^ . O) f4 •" ^ 

G{li^a3a'>^'^53 . 






02 



! 9 



S ..-, ® 'K 



S t^ Q 






<lO 



<« ii M '^ 
2 5 «3 2 

^G ^ S 



r:t< 9* ^ ■C' T*^ T* -^ 

Tj^ Oi 65 Jt^ »b 65 vb CO 

OS (71 r- o CO 1— »— ( tH 



as ib 00 

O rH X^- 



O 






o 
O 



%3 *% ^ 

« in " 

ooW 



00 ir-» 

CO CO 
\0 (M 






rH <yj »p CO 00 

cs lb 00 <i> CO CO <yD tJ^ 

«— iCOCOi— l">!i<Or-((M 



{iH W M <1 <^ <I <1 <1 





. . . . cT 




cT S 'g . 




• . • s s 1 ^ 




•^^^ig^iS 




5 tf bn 5 5sD c3 




- S ^ G 2 G t>C 




C « 5:; § a cj G 


G 


oft] 
of bi 
f bis 
of m 
G of 
of m 
f ma 


■4^ 


U-t 







'T^ 


5-x:2-S.2-33 


;-, 


^ - G ^ >< 


2 


.iiiir-^i 





pq PL, Ph ^ p^ M 



o 



^ 









O 



50 rfi 5i 



IS 
i ^ 

o d 

O ^ 



3 ?^ r^ .. i> ' * * 

S S 2 ^ '^ 1 2 ^ 
■*^ '-> ^ (V .^ <v 



o r^: o 
000 



X X 



. 00 

'^^S'tsOOOGS- 

3 .-■ ^ i^ 1- t^ •-« a> 

c/: pq o Ph Ph Ph M H 



22 



264 



Binary Compounds. 





f^ 




§j 












c 




rs 




^^ 




>i ^ 




h. 




^ P 




;:B ^ 


n 


w^ <A 


'■^ 


>,^ 


•w 


60 t— ( 


^ 


1 ^ 

.1 <5 


E-i 


IP. 




^ H-l 




^^ po 




l^ ^ 




■^U 




-s -^ 




^ w 




-o :^ 




.^o 




S 




Si 




'^ 




<to 




^< 








Jifi 




g 



^^ 



s 



a. 
S 



53 

n3 









^ N 



C' ^ 



O 



OP 



^ g C3 



o 

O G 

X ^ 

c ^ 







<u 


«4-r 






^ 


O 






05 




^ 




O 








;-, 


u 








c3 


Q small d 
11 alcohol 
ammable 




> 


cs 

<D 

5 


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256 Kkcipe roPw Dycing Hats. 



RECIPE FOR DYEING HATS. 

Tlie bath for dyeing hats, employed by the London manufac- 
turers, consists, for 12 dozen, of 

144 Pounds of logwood ; 
12 " green sulpliatc of iron or copperas, 

74- " verdigris. 

The copper is made of a semi-cylindrical shape, and should be 
surrounded with an iron jacket, or case, into which steam may be 
admitted, so as to raise the temperature of the interior bath to 190^ 
Fall., but no higher ; otherwise the heat is apt to affect the stiffen- 
ing varnish, called the gum, with which the body of the hat lias 
been imbued. The logwood having been introduced and digestea 
for some time, the copperas and verdigris are added in successive 
quantities, and in the above proportions, along with every succes- 
sive two or three dozen of hats suspended upon the dipping machine. 
Each set of hats, after being exposed to the bath, with occasional 
airings, during 40 minutes, is taken off the pegs, and laid out upon 
the ground to be more completely blackened by the peroxydize- 
ment of the iron with the atmospheric oxygen. In 3 or 4 hours the 
d3'eing is completed. When fully dyed, the hats are well washed 
in running water. 

A skilful operator furnishes the following valuable information 
•elative to the stiffening of hats. He saj^s : 

All the solutions of gums which I have liitherto seen prepared 
by hatters, have not been perfect, but in a certain degree a mix- 
ture, more or less, of the gums, which are merely suspended, owing 
to the consistency of the composition. When this is thinned by 
the addition of spirit, and allowed to stand, it lets fall a curdy- 
looking sediment, and to this circumstance may be ascribed the 
frequent breaking of hats. My method of proceeding is, first, to 
dissolve the gums, by agitation, in twice the due quantity of spirits, 
whether of wood or wine, and then, after complete solution, draw 
off one half the spirit in a still, so as to bring the stiffening to a 
proper consistency. No sediment subsequently appears on diluting 
this solution, however much it may be done. Both the spirit ajid 
alkali stiffenings for hats made by the following recipes, have been 
tried by some of the first houses in the trade, and have been much 
approved of: 

Spirit Stiffening. — 7 pounds of orange shellac; 2 pounds of gum 
sandarac ; 4 oz. of gum mastic ; \ pound of amber resin ; 1 pint of 
solution of copal ; 1 gallon of spirit of wine, or wood naphtha. 

The shellac, sandarac, mastic, and resin, are dissolved in the 
spirit, and the solution of copal is added last. 

Alkali stiffening. — 7 Pounds of common block shellac ; 1 pound 
of amber resin ; 4 oz. gum thus; 4 oz. gum mastic; 6 oz. borax; 
^ pint of solution of copal. 



Liquefaction of Gases. 



The borax is first dissolved in a little warm water (say 1 gallon) ; 
this alkaline liquor is now put into a copper pan (heated by steam), 
together with the shellac, resin, thus, and mastic, and allowed to 
boil for som.e time, more warm water being added occasionally 
until it is of a proper consistence ; this may be known by pounng a 
little on a cold slab, somewhat inclined, and if the liquor runs off at 
the lower end, it is sufficiently fluid. If, on the contrary, it sets 
before it reaches the bottom, it requires more water. When the 
whole of the gums seem dissolved, half a pint of wood naphtha must 
be introduced, with the solution of copal ; then the liquor must be 
passed through a fine sieve, and it will be perfectly clear and ready 
for use. This stifi^ening is used hot. The hat bodies, before they 
are stifi'ened, should be steeped in a weak solution of soda in water, 
to destroy any acid that may have been left in them (as sulphuric 
acid is used in the making of the bodies). If this is not attended to, 
should the hat body contain any acid Avlien it is dipped into the 
stiff'ening, the alkali is neutralised, and the gums consequently pre- 
cipitated. After the body has been steeped in the alkaline solution, 
it must be perfectly dried in the stove before the stiffening is 
applied ; when stiffened and stovcd, it must be steeped all night in 
water to which a small quantity of the sulphuric acid has been 
added ; this sets the stiffening in the hat body, and finishes the pro- 
cess. A good Avorkman will stiffen 15 or 10 hats a day. If the proof 
is required cheaper, more shellac and resin must be introduced. 

TABLE 

Of Pressures at luhich certain Gases are Liquified, 

Gas :'s t">ic name given to those elastic fluids vdiicli are permanent under a con- 
siderable pressure, and at the temperature zero. 





Lccomes liquid. 










Calculated hoiiing 


jS^ame of Gas. 






point, barometer:^ 
80 inches. 








At 


Under a pressure of 




Sulphurous x\cid, . . 


59 R 


3 atinosi)herc3 


4° Fahr. 


Chlorine, .... 


CO 


4 


22 


Ammonia, .... 


50 


0-5 


04 


Sulphuretted Hydrog. 


50 


IT 


142 


Carbonic Acid, . . 


32 


,"() " 


229 


Hydrochloric Acid, . 


50 


50 


249 


Deutoxide of Azote, . 


45 


50 " 


254 



22* 



258 



Proportionate Strength of Wheels. 






^ 
« 



"to 



=Q 



'I 





g 






i- 

^ ^ 


i J^J 




" 1 


t. (M rS '"' 




«3 

CO 


Q 


1 1 o 2 'fl ^ 




^ 


2^ 


t. per strength F 
sec. in H. P. 

!-27 : 117-60:: 

ickness of cog 
s the pitch, t\ 
3g multiplied 
h. 




X 


co^ 




CO 


(>J 




^ ^ 








Formu 
rength, 


F 

len as 2 

The th] 
I equal 
ss of c< 
e lengt 




OQ 


H ^S:2 




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(N T*< 




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•puooagjad 


00 O 3^ c^ ^ 


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O O lO t- CO i:- CO 

lO Tfi CO (M 1— I OS Ci 




^ ^ ^ ^ ^ 






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vb T^ o <i> ib 

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cp r- CO 3<i CO 


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00 00 OS 4*4 00 


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O»ovoot-Tj<.— (QOOOCOiOOtOOOCOOaOC 


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iO O O O O O lO O O O VO O O O 


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CO CO CO CO 00 


cococ<i<?qs<i;fi'N(?i'^r-('^'^'^'^ 




O Tt< 


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QOfMC^JtOiOCqi-tOsOs-^Ir-OOscO 


UI mpcaie 


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-^ rr" Tt^ CO CO 


C0CCC0C0C0 5<J'M5<I'MG<I.-H— '^^ 




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c^ ^ p qpi^ 


:pocO!?<l'7^pQpr-«pipco^'7«p 




CO CO CO ji iq 


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jbib4j<rj<Tt<Tt<oococococacji5<ic^ 



Knot Table. 



259 















H 
O 






OOOOOOvoOOOOOOOiOOOi— ti0r-(0 
00 CO t^ 1-^ O CO lO O "n^ -T" CO lO CM '>l 1— ( 1— I o O Cs 05 QO 
C^ CM 01 'M C^l (M (M CM CN (M (M CM C<J ^ (>J -ri (M C^l r-i t-H r-i 



2 

CO 


rt<'«*TfiTTi'!^TtiTt"Tj<Tj<-^Tt<Ttirh( 


rj^rt^T^H-Tf^-^T^r^ 


kOCscor-r-iioo-^oocMtrr-^iO 

I— lOOGiCSOOCCir-OtOiOiO"^ 
O <X) «p »p O iO «p O O ip O ip ip 


C5 t}^ OO G<1 CO rH »0 O 
CO CO ^M -M I— 1 r-l O O 
»p >p ip .p p o lO lO 


£ 


OC0OC5(MO00^O00rHTi<00.-(-rt^i:^rH'rJHQ0^^ 

OCiGOr--t-OOiOrt<COCO(Mr-if-i O00i(yj>i--t^?0 
pOiG5O5g5C5CiO5OiOsOiC5CiOiO5C»a)000000QO 


5 


"^?O00OSi— tcOiOlr^CSi— (CCvoi:- 
Or^CO(M«Mr-(Oa50000l:-?OiO 
r^'rJ^TlHrt^'^TH'^OOirSCOCOCOOt) 


crs^cokoir-ai rH-^ 

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O) Oi Oi 05 Oi Oi Ci G5 

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►- .-H rH ,-H (M 

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Qi p p 00 C» 

ko kb kb kb kb 


d CO CO' T^ ri^ kO kO CO 
j:-cOkO-*cO(Mr-iO 
CiOOOQOOOOOqOOOOO 

kb ib kb kb kb kb kb kb 


2 


•-OTtHciast-iocor-i 

CO lO rfi CM ^ O Ci QO 
cpcO"OcOcO<p>pip 
OcOcOcOcOOOO 


05 I:- kc:. CO ^ 

CO .O ^ CO CM 

kp kp kp kp »p 

CO CO CO CO CO 


OiOOCOr^CO— lOQO 
OCTiOOr-COkOTtHCM 
kprl^ri^T^Tj^TJ^r^TT^ 

oi>cbocbcbcbco 


00 


O^OOCOOO(MJ>'int-(McOr-ti:-<M£-(MOOCOa5'<*iO 

OCOCOOCO(MOC5i-OTjHCOr-(OOOr-OTtlCMr-lO 




r-H^OOOOOO 

i-^ iO CO — * Ci r-. o CO 

»p .p >p xp TtH rJ^ r^ -rt^ 
O000GO0000O000C30 


CO r-H (M (M CO 


tJH >0 kO CO 00 Cj O t— 1 


^ CO CO 00 CO 
00 00 00 00 00 


CO q^i (>i c^« c^ q^i (Ti -71 

cooooociociooociooo 


5 


OcTJTt^t-OcOOOCC'OcJiCOt- 
Ol---fi-H05':0C0O00»0'MOr- 
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^kOO'^CiTtiOCO 
kf^ CM 1- rt^ CM as r- 

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r-l 


era era (js GT- us 




OOOOrHCO'^d^cOOOOCOOOO 
OCO(MOO'«*OCOCMQ010^1--C0 

o 05 as 00 oo 00 i— i^ CO CO o kO »p 


.— (iCC»(McOCOkOO 

CO 3^1 Oi kO (M 00 kO 
kO tJi r^ CO 00 CO <M CM 




a 


O CO J:- — t lO 05 CO r- 

O Oi 00 c» I.-- CO CO >0 

kb 4t( 4ti -^ -^ ^ 4f< 4t< 


CO r- o c^ o 

.-H kO O Tt< QO 

>p ^ rj< CO (M 
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OCOOOCOQOCOOCO 
(M t- .-H CO kO ^ 

C>1 -H r-H p p OS p 00 

-^-^Tri-^rt^COCOOO 




CO 


O O O CM Tt< O «C -^ 
O Oi 00 1- CO CO ki^ kO 
P 00 1- CO kp --j^ CO ^ 
OOiOiCSOSOSOSGi 


Ot-i—OOO'MCOr-ll—'^'-'OO 

kO Tt< Tt< -^ kf3 ko lo en CO r- 00 OS o 

7-< P 05 00 J;' CO kp rfi CO <M r-l p p 

asoscbcjooooocjooocjooocjooooo 






0'-<c^c0'<^koc0i:- 


00 Oi O ^ <M 


CCrfikOcOlr-OOasO 




*"* 



260 



Knot Table. 



© 



pq 
< 

H 
O 



s 

r-l 


r-j o ^ :0 r-H t^ CN i^ CM J- c^O CO (^? CO Tt O- lO O lO O O ^ 
GO r^ i-- O CO »0 O Tti Th CC CO (>4 (T'l f— (r-ir— OOCnOiGOQO 


s 

CO 


TtiOOCOl>-C<3coO>OaJTtiGOCOOOC<lJ::-r-ti;OOiOOT+('3i 
OiGOooj::-x:-Ocoio^'*cocO(>4{Mr-H,-(Ooaia5St-- 
-^-^Tt'-^^'^'^'^yi'^'^rr'^Ttirt^'^-^Tj'cccococo 






O00OC0X0ir-O(MTt<t'-O<Mi0t-a»<M'^J:-O<M»0Q0 
QOl— J— <:0O-^rJ^C0CMr-Hr-iO05Q0 1--lr-c0i0OrHC0ff3 



l^-j^-0005OOr-i(MC0'!^iOOJ:^000iO.— i(MC0OcSr-- 
Cr500i:-':ocoOTtHCOC?-ir-toCicX)i:-<:OcoxOTticoc<ir-io 



I— (OO^COi— OTt<C03<JT— (OC5QOX:— cOOCCCNi— i0050D 

■^TtlcOl^rcco<^^<rooO(Wcoc<^G<^c^^(^^(N,(?l 
ococoooo :o«Ci':0':oi>ocO':o<i>. CO. cbi-OOqso 



VOt— lir-(MCOTt^OO(M 05>0i— <j>.r;f4OJ>»C0OC0C0CiC0 



cOT^lr^ooO'-'cOiOir-OiMTHJir-OseMiot-OCOoaiCM 
cOTt^'5<1005j^-iocOf-"000':OrJ^CM»-H05Jt:^cO'rhi(MOOS 
f-H,-H,-M,— (OpOOOOC5a5CiC50i(»(»(» 
cO(»(X)(»00GOa)GO(X)00l~-a:-l^-I^-Ir-l^-I^-i^^^ 



Q0'r+<0i\OO?0C<100'^Oir-C0Or^C0OC0i0fMOr-O 
-*CM05l:^iOG<IOl:^OCOOCO cOCOt— tOiOr+^JMOr-iO 

C;C5O5O5O5O5O5OiOiOiO5C5CiO5C5O5O5C5C5(j500G0 



rJ^OiOi— ilr-COCsiC'MCitDCOOOO'rt^^Oqr-iOiGOt-O 
(^li-H,— ir-nOpOOSC^CSQOGOOOI^r-r-COQDOiOipJO 



COOCOO'rt<COCO!MCOCO'^»Ci:OQO— 'CCCDOCOX^i— lO 
OlTfOOCOGOCOGOcOCOCCQOCOOOCOOi'rJ-CJV-OOiOi— CO 

^J-^ooovOT^'Ttl(^5l^(^^(^^7-'7-pp9»Oi9iooqoJ^^ 

W. CC (^COCOOtCO(^0OiWi^O5lOCO(WMC^C^ 



OCCTt^r^OiO--r^iOCO— <^— ^(Mrpj^OrJ^QOCOOSCO 
.— I'MCO'+iOj^OSO'n'rtitOGCOCMTt^OOSr-iCOCOQOi— t 
O <» r- CO O rti <^ <^:> (>1 'T- p Ci C5 (» r- o o ip -;;t^ 



Knot Table. 



261 






I 

s 

H 
O 





i:- <M 1- CO 00 'rf^ OS -rH O lO ^ to C'l t- CC 00 -f< 

ooocpooooooooooppo 

1 1 


B 

CO 

t-t 


Tt<QOCCOOCOt-(Mt-(M<:OrHCOr-H«6.-HOO 

i:-cO<:DOiOT}<'«*iCOCOC^1iM.-ir-iOOaiai 
cc CO CO CO CO JO CO CO o:) CO oo ot> CO CO CO 5<> c<i 




oOG<i«oocoi:-.-H»oo5cor-'-HiOCicoi^-r-i 

r-<r-tOOC:5Q0Q0£^O'"X>kOV0TtlC0C0<MCM 
"vj* M* ^r ^7^ ^^ ^7^ ^41 ^T^ ^g^ '^J^ ^^^ ^5^ ^^ ^7^ ^"^ ^5^ ''^ 


^ 


.-•Tf<:00ii--<Tfi£-OC0C005CM»0Q0CC0C0 ! 
(Mr— (OC50:>COj>'i::--':OOrti-rHCOC'13^i— lO 
T'T'T'PP99POOOOppOOp 


1 

C5 


QOO«-"C<JTt<iOir-aoO'-icO'^<:0£-057-Hco 

OsCiOOr-oo-rt^cococ^lr-iOOscoxr-xr-O 
»o >^o ip »^o »^p ip ip ip ip Tti T^i tH 7i^ Tt< 


Tt^T:t^COCO'^5<^^!M-^,-l,-^rHOOOOOO 

ir- '-0 n< CO <M T-l C5 GO 1-- iO -* 0:1 'M .-< 

•r'':^r'T'T"T'T'':^^99P99999 


(» 

2 
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COO.t-'^i-HQOiOClCitO'Tff.-jaitOTit^^O 
OOl-^iOrt^CO'— lOOSt^OiO-^C^J.— lOCSh- 

?C«0«DtO?COcr)«OCC<:0<X>OCO:0000 


10 00 1-H iO QO 'M 10 Ci CO ir- !-( vo Ci CO Jr- i-x 
i- <0 Tt< S<I 05 !■- 10 Tt^ CM '-^ C5 i— to Tfl CO — 
^ r- 1;- J,^ i- to '>p p CO <p -X; kp »p ip kp kp »p 




CO»-'C5£-OCO— ^OC5t-';0>0>OT+<C0 5^1.-l 
COr— lODO-^'MOOOkCCO.— "O^C^kOCOrHwi 

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262 Cements. 



C E M E is^ T S . 

Shell-lac Cement, or Liquid Glue. — Fine orange sliell lac, bruised, 4 
oz. ; liighly rectified spirit, 3 oz. Digest in a warm place, frequent- 
ly shaking, till the shell-lac is dissolved. Rectified wood naplitha 
may be substituted for spirit of wine, where the smell is not 
objectionable. This is a most useful cement for joining almost any 
material. 

Shell-lac Cement, vntJiout Spirit. — Boil 1 oz. of borax in 16 oz. 
water; add 2 oz. powdered shell-lac, and boil in a covered vessel 
till the lac is dissolved This is cheaper than the above, and for 
many purposes, answers very Avell. Both are useful in fixing paper 
labels to tin, and to glass when exposed to damp, 

Keller s Armenian Cement, for Gla^s, China, d'c. — ^Soak 2 dr. of 
cut isinglass in 2 oz. of water for 24 hours; boil to 1 oz. ; add 1 oz. 
of spii-it of wine, and strain through linen. Mix this, wiiile hot, 
with a solution of 1 dr. of mastic in 1 oz. of rectified spirit, and 
triturate with ^ dr. powdered gum ammoniac, till perfectly homo- 
geneous. 

Di\ Ure's Diamond Cement. — Isinglass, 1 oz. ; distilled water, 
6 oz. ; boil to 3 oz., and add 1^ oz. of rectified spirit. Boil for a 
minute or two, strain, and add, while hot, first, -^ oz. of a milky 
emulsion of ammoniac, and then 5 dr. of tincture of mastic. 

Hoenles Cement, for Glass or Earthenware. — Shell-lac, 2 parts; 
Venice turpentine, I part. Fuse together, and form into sticks. 

Cheese Cement, for Earih^riware, dbc — Mix together white of egg, 
beaten to a froth, quick-lime, and grated cheese. Beat them to a 
paste, which forms an excellent cement. 

Cui'd Cement. — Add ^ pint of vinegar to 4- pint of skimmed milk. 
Mix the curd with the wdiites of 5 eggs well beaten, and sufficient 
powdered quick-lime to form a paste. It resists water, and a mode- 
rate degree of heat. 

Cement for joining Spar and Marble Ornaments, d;c. — Melt toge- 
ther 8 parts of resin, 1 of wax, and stir in 4 parts, or as much as 
may be required, of Paris plaster. The ])ieces to be made hot. 
_ Henslers. Cement. —Grvin^ 8 parts of litharsje, 2 of recently burnt 
lime, and 1 of white bole, with linseed oil varnish. Tliis is a veiy 
tenacious cement, but it takes considerable time to dry. 

Si7irjcr's Cement, for Electric^d Machines and Galvanic Troughs. — 
Melt together 5 lbs. of resin, and 1 lb. of beeswax, and stir in 1 lb 
of red ochre (hicrhlv dried, and still warm), and 4 oz. of Paris plas- 
ter, contin-uinsr the heat a little above 212°, and stirring constantly 
till all frothing ceases. Or (for troughs), resin, 6 lbs ; dried red 
ochre, 1 lb. ; calcined plaster of Paris, | lb. ; linseed oil, ^ lb. 



Cements. 



263 



Composition for welding Cast Steel, — Take of borax, 10 parts, sal 
ammoniac, 1 part; grind or pound them roughly together ; then 
fuse them in a metal pot over a clear fire, taking care to continue 
the heat until all spume has disappeared from the surface. When 
the liquid appears clear, the composition is ready to be poured out 
to cool and concrete ; afterwards, being ground to a fine powder, it 
is ready for use. * * -^ To use this composition • The steel to 
be welded is first raised to a "bright yellow" heat, it is then 
dipped among the welding powder, and again placed in the fire, 
until it attains the same degree of heat as before ; it is then ready 
to be placed under the hammer. 

Cast-Iron Cement. — Take of clean iron borings, or turnings, 1 
cwt. ; of sal-ammoniac 8 oz. ; and 1 oz. of flour of sulphur. Mix 
them thoroughly, and add sufficient water. If the cement is not to 
be immediately used, care should be taken to keep the mixture 
soaked in water; if left dry, the cement will heat, and be spoiled. 

Cement for Steam Pipe Joints^ (be., ivith Faced Flanges — To 2 
parts of white lead mixed, add 1 part of red lead dry , grind, or 
otherwise mix them, to a consistence of thin putty ; appl}^ inter- 
posed layers, with one or two thicknesses of canvas or gauze wire, 
as the necessity of the case may require. 

Glues. — 1. A very strong glue is formed by throwing a small 
quantity of powdered chalk into melted common glue. 

2. To'^ make a glue which will resist the action of w^ater — boil 
one pound of common glue in two quarts of skimmed milk. 

Botany Bay Cement. — Take 1 part of Botany Bay gum, and melt 
and mix it with 1 part of brickdust. 

Cap Cement. — As Singer's; but 1 pound of dried Venetian red 
may be substituted for the red ochre and Paris plaster. 

Bottle Cement. — Resin 15 parts ; tallow 4 (or wax 8) parts ; highly 
dried red ochre 5 parts. The common kinds of sealing-wax are 
also used. 

Turner's Cement. — Beeswax 1 oz. ; resin | oz. ; pitch I- oz. Melt, 
and stir in fine brickdust. 

Coppersmith's Cement— ?owdiQVQ^ quick-lime, mixed with bul- 
lock's blood, and applied immediately. 

Engineers' Cement. — Equnl weights of red and white lead, with 
drying oil, spread on tow or canvas. This is an admirable compo- 
sition for uniting large stones in cisterns, (fee. 

Iron Cement for Closing the Joints of Iron Pipes. — Take of iron 
borings, coarsely powdered, 5 lbs. ; of powdered sabammoniac 2 
oz. ; of sulphur 1 oz. ; and water sufficient to moisten it. Tliis com- 
position hardens rapidly ; but if time can be allowed it sets more 
firmly without the sulphur. It must be used as soon as mixed, and 
rammed tightly into the joints. 



264 Cements. 

Cement for Steam Pipes. — Good linseed oil varnish ground, with 
equal weights of white lead, oxide of manganese, and pipeclay. 

Gad's Hydraulic Cement. — Powdered clay 3 lbs. ; oxide of iron 
1 lb ; and boiled oil to form a stiff paste. 

Cements for Masonry of Chambers of Chlorine^ &c. — Equal parts 
of pitch, rosin, and plaster of Paris. 

Roman Cement. — 1 bushel of slacked lime; 8^ lbs. of green cop-f 
peras ; and ^ bushel of fine gravel sand. The copperas should be f 
dissolved in hot water. It must be stirred with a stick, and kept ' 
stirred continually while in use. Care should be taken to mix at. j 
once as much as may be requisite for one entire front, as it is very ^ 
difficult to obtain the same shade or color a second time. It ought 
to be mixed the same day it is used. This is the English Koman 
cement. j 

The genuine Roman cement consists of \^aQ pulvis puteolanus, or \ 
puzzolene, a ferruginous clay from Puteoli, calcined by the fires of 
Vesuvius, lime, and sand, mixed with soft water. The only prepa- 
ration which the puzzolene undergoes is that of pounding and 
sifting ; but the ingredients are occasionally mixed with bullock's 
blood and suet, to give the composition greater tenacity. 

Seal Engravers' Cement. — Resin 1 part ; brickdust 1 part. Mix, 
with heat. 

Marine Cement, commonly called Marine Glue. — Cut caoutchouc 
into small pieces, and dissqlve it, by heat and agitation, in coal 
naphtha. Add to this solution powdered shell-lac, and heat the 
whole, with constant stirring, until combination takes place ; then 
pour it, while liot, on metal plates, to form .sheets. When used, it 
must be heated to 280° Fah., and applied with a brush. 

Liquid Glue. — Dissolve bruised orange shell-lac in \ of its weight 
of rectified spirit, or of rectified wood naphtha, by a gentle heat. It 
is very useful as a general cement and substitute for glue. Another 
kind may be made by dissolving 1 oz. of borax in 12 oz. of soft 
water, adding 2 oz. of bruised shell-lac, and boiling till dissolved, 
stirring it constantly. 

Banlc Note Glue. — Dissolve 1 lb. of fine glue, or gelatine, in 
! water ; evaporate it till most of the water is expelled ; add -^ lb. of 
, brown sugar, and pour it into moulds. Some add a little lemon 
I juice. It is also made with 2 parts of dextrine, 2 of water, and 1 j 
I of spirit. j 

MaissiaCs Cement, as an Air-Tight Covering for Bottle.% <kc. — i 
Melt india-rubber (to which 15 per cent, of wax or tallow may be! 
added), and gradually add finely powdered quick-lime, till a change I 
of odor shows that combination has taken place, and a proper con- 
sistence is obtained. [ 

Cement for Attaching Metal Letters on Plate Glass. — Copal vai'nish | 
15 parts; drying oil 5 parts; turpentine 3 parts; oil of turpentine! 



Cements. 265 

j 
2 parts ; liquified glue 5 parts. Melt in a water bath, and add 10 
parts of slacked lime. 

Japanese Cement — Mix rice flour intimately with cold water, and. 
boil gently. 

French Cement. — Mix thick mucilage of gum arable with pow- 
dered starch. 

Stone Cement. — River sand 20 parts; litharge 2 parts ; quick-lime 
1 part. Mix, with linseed oil. 

Plambers Cement. — Resin 1 part; brick-dust 2 parts. Mix, with 
heat. 

Paruia7i {Jement. — Gum arable 1 oz. ; water 2 oz. ; suflicient 
starch to thicken. 

Cnnent for Floors. — The following style of floor is well adapted 
for plain country dwellings: Take two thirds of lime, and one of 
coal ashes, well sifted, with a small quantity of loam clay; mix the 
whole together, temper it well with water, andTmake it up into a 
heap; let it 113 six or seven days, and then temper it again. After 
this, heap it up for three or four days, and repeat the tempering! 
very high, till it becomes smooth, yielding, tough, and gluey. The i 
gi-onnd being then levelled, lay the floor therewith about 2|- or 3 i 
inches thick, making it smooth with a trowel. The hotter the! 
season is the better ; when thoroughly dried it makes a capital i 
floor. Should a better looki7uf floor be desired, take lime of rag! 
stones, well tempered with white of eggs, and cover the floor half an , 
inch thick with it, before the under flooring is too dry. If this be; 
well done, and the floor thoroughly dried, it will look, when! 
rubbed with a little oil, as transparent as metal, or glass. I 

Common Paste. — To a table-spoonful of flour add gradually half ai 
pint of cold water, and mix till quite smooth ; add a pinch of pow- 
dered alum (some add a small pinch of powdered rosin), and boil 
for a few moments, stirring constantly. The addition of a little 
brown sugar, and a few grains of corrosive sublimate, will, it is 
said by practical chemists, preserve it for years. 

Soft Cement. — Melt yellow wax with half its weight of comnion 
turpentine, and stir in a little Venetian red, previously well dried 
and finely powdered. This cement does very well as temporary 
stopping for joints and openings m glass and other apparatus, 
where the heat and pressure are not great. 

Lutes, or Cements, for Closincr the Joints of Apparatics. — Mix 
Paris plaster witli water to a soft paste, and apply it immediately. 
It bears nearly a red heat. It may be rendered impervious b}' 
rubbing it over with wax and oil. 

^7<o^/<<?r.— Slacked lime, made into a paste with white of egg, or a 
solution of gelatine. 

Another. Fat Lute. — Finely powdered clay, moistened with 
water, and beaten up with boiled linseed oil. Roll it into cyhnders. 



266 Cements. 

j ~ 

'and press it on the joints of the vessels, which must be perfectly 
dry. It is rendered more secure by binding it with strips of linen 
moistened with white of egg. 

Another. — Linseed meal beaten to a paste with water. 

Another. — ^Sljps of moistened bladder, smeared with white of 

Fire and Waterproof Cement. — To half a pint of milk put an 
equal quantity of vinegar, in order to curdle it ; then separate the 
curd from the whey, and mix the latter with four or five eges, 
beating the whole well together. When it is well mixed adcTa 
little lime through a sieve, until it has acquired the consistence of a 
thick paste. With this cement broken vessels may be united. It 
1 resist-s water, and, to a certain extent, fire. 

I Fire LuieR. — The following composition will enable glass vessels 
ito sustain an incredible degree of heat: Take fragments of porce- 
lain, pulverize, and sift them well, and add an equal quantity of 
fine clay, previously softened with as much of a saturated solution 
of muriate of soda as is requisite to give the whole a proper con- 
sistenca Apply a thin and uniform coat of this composition to the 
glass vessels, and allow it to dry slowly before they are put into 
the fire. 

Another. — Equal parts of coarse and refractory clay, mixed with 
a little hair, form a good lute. 

A Cement for Stopping the Fissures of Iron Vessels. — Take two 
ounces of muriate of ammonia, 1 ounce of flour of sulphur, and 16 
ounees of cast-iron filings, or turnings. Mix them well in a mortar, 
and keep the powder dry. When the cement is wanted take one 
part of this and twenty parts of clean iron filings, or borings; grind 
them together in a mortar, mix them \\ith water to a proper con- 
sistence, and apply them between the joints. This cement answers 
for flanges of pipes, <fcc., about steam-engines. 

Genuine Armenian Cement. — '*' The jewellers of Turkey, who are 
mostly Armenians," says Mr. Eton, a very intelligent traveller, and 
I at one time a resident and consul in that country, "have a singular 
j method of ornamentinor watch cases, drc, with diamonds and other 
I precious stones, by simplv glueing or cementing them on. The 
I stone is set in silver or gold, and the lower part of the metal made 
I flat, or to correspond with the part to which it is to be fixed. It i 
:is then warmed gentiv, and the ^"lue applied, which is so very | 
; strong that the parts' thus cemented never separate. This glue, j 
■which will firmly unite bii^ of glass, and even polished steel, and [ 
:may of course be applied to a vast variety of useful purposes, is | 
;thus made: — Dissolve five or six bits of gum mastic, each the size | 
of a large pea, in as much spirits of wine as will suffice to render it 
'liquid; in another vessel dissolve as much isinglass, previously a 
'little softened in water (thouofh none of the water must be used), in 
I French brandy, or good rum, as will make a two ounce phial of 



Cements. 267 

very strong glue, adding two small bits of gum galbaniim, or ammo- 
niacum, which must be rubbed or ground till they are dissolved. 
Then mix the whole with a sufficient heat, keep the glue in a phial 
closely stopped, and when it is to be used set the phial in boiling 
water." 

AnoUier. — Thick isinglass glue 1 part ; thick mastic varnish 1 
part. Melt the glue, mix, and keep it in a closely corked phial. 
For use, put the phial in hot water. 

Elantic Cement for Belh, — Dissolve in good brandy a sufficient! 
quantity of isinglass, so as to be as thick as molasses. 

A very strong Carpenters^ Glue. — Dissolve an ounce of the best ! 
isinglass, witli a moderate heat, in a pint of water. Take this solu-j 
tion, and sti*ain it through a piece of cloth, and add to it a propor-| 
tionate quantity of the best glue, Avhich has been previously soaked 1 
for about twent^'-four hours, and a gill of vinegar. After the whole | 
of the materials have been brought into a solution, let it once boil I 
up, and strain off the impurities. This glue is well adapted for any; 
work which requires particular strength, and where the joints them- j 
selves cPo not contribute towards the combination of the work ; or | 
in small fillets and mouldings, and carved patera, that are held on j 
the surface by the glue. i 

A Olue for Inlayiruf Brass or Silver Strings, d:c. — Melt your glue i 
as usual, and to every pint add of tinely powdered rosin and finely! 
powdered brickdust two spoonfuls each; incorporate the whole I 
together, and it will hold the metal much faster than any common j 
glue. 

A strong Glue that will resist Moisture. — Dissolve gum sandarac 
and mastic, of each ^ of an ounce, in ^ oi a pint of spirit of wine, to 
which add ^ of an ounce of clear turpentine. Now take strong 
glue, or that in which isinglass has been dissolved ; then, putting 
the gums into a double glue-pot, add by degrees tlie glue, con- 
stantly stirring it over the fire till the whole is well mixed ; then 
strain it through a cloth, and it is ready for use. You may now 
return it into the glue pot, and add ^ an ounce of very finely pow- 
dered glass ; use it quite hot. If you join tAvo pieces of wood toge- 
ther with it you may, when perfectly hard and dry, immerse it in 
water and the joint will not separate. 

A Paste for laying Cloth or Leather on Table Tops. — ^To a pint of 
the best wheaten flour add two table spoonfuls of finely powdered 
rosin, and one spoonful of powdered alum. Mix them well together, 
put them into a pan, and add by degrees rain water, carefully 
stirring it till it is of the consistence of thinnish cream ; put it into 
a saucepan over a clear fire, keeping it constantly stirred, tliat it 
may not get lumpy. When it is of a stiff consistence, so that the 
spoon will stand upright in it, it is done enough. Be careful to stir 
it well from the bottom, for it will burn if not well attended to. 
Empty it out into a pan, and cover it over till cold, to prevent a 



268 



Cements. 



skin forming on the top, which would make it himp^-. This paste 
is very superior for tlie purpose, and adhesive. To use it for cloth ', 
or baize spread tlie paste evenly and smoothly on the top of the ; 
table, and lay your cloth on it, pressing and smoothing it with a ' 
flat piece of wood ; let it remain till dry; then trim the edges close I 
to the cross-banding. If you cut it close at first it will, in drying, | 
shrink and look bad where it meets the banding all round. If used I 
for leather, the leather must be first previously dampened, and the | 
paste then spread over it; then lay it on the table, and rub it' 
smooth and level with a liueu cloth, and cut the edges close to the 
banding with a shoi-t knife. Some lay their table-cover with glue 
instead of paste, and for cloth perhaps it is the best method ; but 
for leather it is not proper, as glue is apt to run through. In using 
it for cloth, great care must be taken that your glue is not too thin, 
and that vou rub the cloth well down with a thick piece of wood 
made hot*^ at the fire, for the glue soon chills. You ma}' by this 
method cut oft' the edges close to the border at once. 

Cement Stoppinr/. — Mix equal quantities of sawdust, of the same 
wood required to be stopped, and clear glue; and with t^^is stop 
up the holes or defects of the wood. Where the surface is to be 
japanned or painted, whiting may be used instead of sawdust. Be 
sure to let the stopping dry before jou attempt to finish the surface. 

Mahogany-colored Cement. — -Melt two ounces of beeswax, and 
half an ounce of rosin, together; then add half an ounce of Indian 
red, and a small quantity of yellow ochre to bring the cement to 
the desired color. Keep it in a pipkin for use. 

A Cement to stop Maws or Cracks in Wood of any Color. — Put 
any quantity of fine sawdust, of the same wood your work is made 
with, into an earthen pan, and pour boiling water on it, stir iti 
well, and let it remain for a week or ten days, occasionally stirring | 
it; then boil it for some time, and it will be of the consistence of! 
pulp or paste ; put it into a coarse cloth, and squeeze all the mois- 
ture from it. Keep for use, and when wanted mix a sufficient 
quantity of thin glue to make it into a paste ; rub it well into the 
cracks, or fill up the holes in your work with it. When quite hard 
and dry, clean your work oiff, and, if carefully done, you will 
scarcely discover the imperfection. 

Fireproof Stucco for Wood, dx. — Take moist gravelly earth (pre- 
viously washed), and make it into stucco with the following com- 
position : Pearlashes two parts; water five parts; common clay 
one part. It has been tried on a large scale and found to answer. 

Terra Cotta — Potter's clay, Ryegate sand, and water, each a suf- 
ficient quantity. Model and bake. 

Peivs Composition for covering Buildings. — Take the hardest and 
purest limestone (white marble is to be preferred), free from sand, 
clay, or other matter ; calcine it in a reverberatory furnace, pulverize 
and pass it through a sieve. One part, by weight, is to be mixed 



Analysis of certain Organic Substances. 



269 



with two parts of clay well baked and similarly pulverized, con- 
ducting the whole operation with great care. This forms the first 
powder. The second is to be made of one part of calcined and pul- 
verized gypsum, to which is added two parts of clay, baked and 
pulverized. These two powders are to be combined, and intimately 
mcorporated, so as to form a perfect mixture. When it is to be 
used, mix it with about a fourth part of its weight of water, added 
gradually, stirring the mass well the whole time, until it forms a 
thick paste, in which state it is to be spread like mortar upon the 
desired surface. It becomes in time as hard as stone, allows no 
moisture to penetrate, and is not cracked by heat. When well 
prepared it will last any length of time. When in its plastic or 
soft state, it may be colored of any desired tint. 



TABLE 

Of Analysis of certain Organic Suhstavcf.\ from the best 
authorities. 





Carbon. 


Hydrogen. 


Oxygen. 


Nitrogen. 


Total. 


Sugar, .... 


42-225 


6-600 


51-175 


_ 


100 


Starch, . 






44-250 


6-674 


49 076 


— 


100 


Gum, . . . 






42-682 


6-374 


50-944 


— 


100 


Lignin, 






52-53 


5-69 


41-78 


— 


100 


Tannin, . 






52-590 


3-825 


43-585 


— 


100 


Indigo, 






73-260 


2-500 


10-43 


13-81 


100 


Camphor, 






'73-38 


10-67 


14-61 


-34 


100 


Caoutchouc, 






87-2 


12-8 


— 


— 


100 


Albumen, 






52-883 


7-540 


23-872 


15-705 


100 


Fibrin, 






53-36 


7-021 


19-685 


19934 


100 


Casein, . 






59-781 


7-429 


11-409 


21-381 


100 


Urea, . . 






18-9 


9-7 


26-2 


45-2 


100 


Gelatine, . 






47-881 


7-914 


27 207 


16-998 


100 


Picromel, 






54-53 


1-82 


43-65 


— 


100 


Hordein, . 






44-2 


6-4 


47-6 


1-8 


100 


Emelin, . 






64-57 


7-77 


22-95 


4-3 


100 


Yeratrin, . 






66-75 


8-54 


19-60 


5-04 


100 


Cinchonin, 






77-81 


7-37 


5-93 


8-89 


100 


Quinin, . 






75-76 


7-52 


8-61 


811 


100 


Brucin, . 






70-88 


6-66 


1739 


5-07 


100 


Strychnin, 






76-43 


6-70 


11-06 


5-81 


100 


Narcotin, 






65-00 


5-50 


26-99 


251 


100 


Morphin,. 






72-340 


6-366 


16-299 


4-995 


100 



23* 



2'70 



Toothed Wheels. 



TABLE ! 

To Calculate the Pitch of a Toothed Wheel, when the radius and 
number of teeth are given ; and the radius, when the pitch and 
number of teeth are given, from 10 to 159 teeth. 



^5 




o5 




"Sf 




! ^5 




'35 


oS 


Radius. 


6l 


Radius. 


6% 


Radius. 


1 . QJ 
, O !U 


Radius. 


oS 


^£^ 




^H 




ZH 




'i^r^ 




I^H 


10 


1-618 


40 


6-373 


70 


11144 


100 


15-918 


130 


11 


1-774 


41 


6-532 


71 


11-303 


101 


16-077 


131 


12 


1-932 


42 


6-691 


72 


11-463 


102 


16-236 


132 


13 


2-089 


43 


6-850 


73 


11-622 


103 


16-395 


133 


14 


2-247 


44 


7-009 


74 


11-781 


104 


16-554 


134 


15 


2-405 


45 


7-168 


75 


11-940 


105 


16-713 


135 


16 


2-563 


46 


7-827 


76 


12-099 


106 


16-873 


136 


17 


2-721 


47 


7-486 


77 


12-258 


107 


17-032 


137 


18 


2-879 


48 


7-645 


78 


12-417 


108 


17-191 


138 


19 


3-038 


49 


7-804 


79 


12-576 


109 


17-350 


139 


20 


3-196 


50 


7-963 


80 


12-735 


110 


17-509 


140 


21 


3-355 


51 


8-122 


81 


12-895 


111 


17-668 


141 


22 


3-513 


52 


8 281 


82 


13-054 


112 


17-827 


142 


23 


3-672 


53 


8-440 


83 


13-213 


113 


17-987 


143 


24 


3-830 


54 


8-599 


84 


13-370 


114 


18-146 


144 


25 


3-989 


55 


8 758 


85 


13-531 


115 


18-305 


145 


26 


4-148 


56 


8-917 


86 


13-690 


116 


18-464 


146 


27 


4-307 


57 


9-076 


87 


13-849 


117 


18-623 


147 


28 


4-465 


58 


9-235 


88 


14-008 


118 


18-782 


148 


29 


4-624 


59 


9-394 


89 


14-168 


119 


18-941 


149 


30 


4788 


60 


9-553 


90 


14-327 


120 


19-101 


150 


31 


4-942 


61 


9-712 


91- 


14-486 


121 


19-260 


151 


32 


5-101 


62 


9-872 


92 


14645 


122 


19-419 


152 


33 


5-260 


63 


10-031 


93 


14-804 


123 


19-578 


153 


34 


5-419 


64 


10-190 


94 


14-963 


124 


19-737 


154 


35 


5-578 


65 


10-349 


95 


15-122 


125 


19896 


155 


36 


5737 


66 


10 508 


96 


15-281 


126 


20055 


156 


37 


5-896 


67 


10-667 


97 


15-440 


127 


20-214 


157 


38 


6055 


68 


10-826 


98 


15-600 


128 


20-374 


158 


39 


6-214 


69 


10-985 


99 


15-759 


129 


20-533 


159 



Radius. 



20-692 
20-851 
21-010 
2M69 
21-328 
21-488 
21-647 
21-806 
21-965 
22-124 
22-283 
22-442 
22-602 
22-761 
22-920 
23 079 
23-238 
23 397 
23-556 
23-716 
23-875 
24034 
24-193 
24-352 
24-511 
24-620 
24-830 
24-989 
25-148 
25-307 



Rule 1. — Divide the required radius by the radius opposite the 
given number of teeth in the table; the quotient will be the 
required pitch of the wheel. 

Example. To find the pitch of a wheel whose radius is 43 
inches, that shall contain 90 teeth : 

Required radius 43 -i- 14-327 = 3-inch pitch. 

Rule 2. — Multiply the radius opposite the given number of 



Cables. 



271 



teeth in the table, by the pitch required ; the product will be the 
required radius of the wheel. 

Example. To find the radius of a wheel that shall contain 48 
teeth of 2^ -inch pitch : 

In the Table, radius 7*645 x 2*5 = ^^jq inches nearly. 







CABLES. 












TABLE 








For finding the Sir a 


ji that may safely be applied to a 
Cable. " 


good Hempen 


Circum. 


Pounds. 


Circumfer. 


Pounds. 


1 

i Circmmfer. 


Pounds. 


6- 


4320- 


10*25 


12607-5 


^ 14-50 


25230* 


6-25 


46B7-5 


10-50 


13230- 


14-75 


26107*5 


6-50 


5070- 


10-75 


13867-5 


15- 


27000* 


6-75 


5467-5 


11* 


14520* 


15-25 


27907*5 


7- 


5880- 


11-25 


15187-5 


15-50 


28830* 


7-25 


6307 • 5 


11-50 


15870- 


i 15-75 


29767*5 


7-50 


6750- 


11-75 


16567-5 


16- 


30720- 


7-75 


7207-5 


12- 


17280- 


1 16-25 


31687-5 


8- 


7680- 


12-25 


18007*5 


16-50 


32 


670- 


8-25 


8167-5 


12-50 


18750* 


16-75 


33 


667*5 


8-50 


8670- 


12*75 


19507*5 


17- 


34 


680* 


8-75 


9187-5 


IS- 


20280- 


17*25 


35 


707-5 


9- 


9720- 


IS -25 


21067-5 


17-50 


36750* 


9-25 


10267-5 


13-50 


21870- 


17-75 


37807*5 


9-50 


10830- 


13-75 


22687-5 


18- 


38880* 


9-75 


11407*5 


14- 


23520- 


18-25 


39967*5 


10- 


12000- 


14-25 


24367-5 







To ascertain the Strength of Cables. — Multiply the square of the j 
circumference in inches by 120, and the product is the weight the 
cable will bear in pounds, with safety. 

To ascertain the Strength of Ropes. — Multiply the square of the 
circumference in inches by 200, and it gives the weight the rope 
will bear in pounds, with safety. 

To ascertain the Weight of Manilla Ropes and Hawsers. — Multiply 
the square of the circumference in inches by -03, and the product is 
the weight in pounds of a foot in length. 

This is but an approximation, sufficiently correct for many pur- 
poses. • 



21% 






Cables 


AND Anchors. 










TABLE 






Showing the Size of 


Cables and Anchors proportional to the 
of Vessels. 


Tonnage 


Tonnage of 
vessels. 


Cables. 
Circutnfer. 
in inches. 


Chain Ca- 
bles. Diam. 
in inches. 


Proof 

in 
tons. 


Weight of 
Anchor in 
pounds. 


Weight of 

a fathom of 

chain. 


Weight of 

a fathom ot 

Cable. 


5 


3 




■i% 


•f 


56 


5-i 


2-1 


8 


4 




•1 


1-f 


84 


8- 


4- 


10 


4 


'i 


■i\ 


2-i 


112 


11- 


4-6 


15 


5 


'i 


•i 


4- 


168 


14- 


6-5 


25 


6 




•^ 


5- 


224 


17- 


8-4 


40 


6 


'i 


•f 


6- 


336 


24- 


9-8 


60 


1 




■\i 


7- 


392 


27- 


11-4 


15 


1 


i 


•f 


9- 


532 


30- 


13- 


100 


8 




•H 


10- 


616 


36- 


15- 


130 


9 




•1 


12- 


700 


42- 


18-9 


150 


9 


i 


■H 


14- 


840 


50- 


21- 


180 


10 


i 


1- 


16- 


962 


56- 


25-7 


200 


11 




i-iV 


18- 


1176 


60- 


28-2 


240 


12 




i-i 


20- 


1400 


70- 


33-6 


270 


12 


i 


i-tV 


21- 


1456 


78- 


36-4 


320 


13 


i 


1-i 


22-i 


1680 


86- 


42-5 


360 


14 




1-A 


25- 


1904 


96- 


45-7 


400 


14 


i 


1-f 


27- 


2072 


104- 


49- 


440 


15 


i 


1-tV 


30- 


2240 


115- 


56- 


480 


16 




i-i 


33- 


2408 


125- 


59-5 


520 


16 


i 


i-rV 


36- 


2800 


136- 


63-4 


570 


17 




i-i 


39- 


3360 


144- 


67-2 * 


620 


17 


1 


i-H 


42- 


3920 


152- 


71-1 


680 


18 




i-f 


45- 


4200 


161- 


75-6 


740 


19 




i-H 


49- 


4480 


172* 


84-2 


820 


20 




1-1 


52- 


5600 


184- 


93-3 


900 


22- 




i-H 


66- 


6720 


196- 


112-9 


1000 


24 




1- 


60- 


7168 


208- 


134-6 



Hempen Rope — Copper Rods. 



273 



TABLE 

For finding the Strain that may he applied to a Hempen Rope with 

safety. 



Circum. 


Pounds. 


Circumfer. 


Pounds. 


Circumfer. 


Pounds. 


1' 


200- 


3-50 


2450* 


6- 


7200- 


1-25 


312 


5 


3-75 


2812-5 


6-25 


7812-5 


1-50 


450 




4- 


3200- 


6-50 


8450- 


1-75 


612 


5 


4-25 


3612-5 


6-75 


9112-5 


2- 


800 




4-50 


4050- 


7- 


9800- 


2-25 


1012 


5 


4-75 


4512-5 


7-25 


10512-5 


2-50 


1250 




5- 


5000- 


7-50 


11250- 


2-75 


1512 


5 


5-25 


5512-5 


7-75 


12012'5 


3- 


1800 




5-50 


6050- 


B- 


12800' 


3-25 


2112-5 


5-75 


6612-5 







TABLE 

Of Weight of Copper Rods or Bolts, from i to 4: inches in diameter, 
and 1 foot in length. 



Diam 


Pounds. 


Diamete 


r. Pounds. 


Diameter. 


Pounds. 


T 


•1892 


i-i 


3-8312 


2-1 


17-0750 


. 5 

r6- 


-2956 


1. 3 
i TF 


4-2688 


2i 


18-9161 


•f 


-4256 


li 


4-7298 


2-i 


20-8562 


. 7 


•5794 


1-/^ 


5-2140 


2-f 


22-8913 


•i 


-7567 


li 


5-7228 


n 


25-0188 


■h 


•9578 


l-r's 


6-2547 


3- 


27-2435 


•i 


1-1824 


li 


6-8109 


s-i 


29-5594 


.1 1 


1-4307 


1-A 


7-3898 


3i 


31-9722 


•1 


1-7027 


l-f 


7-9931 


3-f 


34-4815 


.13 
fF 


1-9982 


l-i 


9-2702 


3-1 


37-0808 


i 


2-3176 


l-J 


10-6420 


s-f 


39-7774 


•H 


2-6605 


2- 


12-1082 


3-f 


42-5680 


1- 


3-0270 


2^- 


13-6677 


n 


45-4550 


i-tV 


3-4170 


2-i 


15-3251 


4- 


48-4330 





274: 



Copper Pipes — Cisterns. 



Weight of a copper rod 12 inches long and 1 in. diameter = 
3-039 lbs. 

AVeiglit of a brass rod 12 inches long and 1 inch diameter = 2"86 
lbs. 

TABLE 

Of the Weight of Riveted Copper Pipe^, from 5 to 80 inches in 

dlaiaeter, from 3 to ^^ thick, and 1 foot in length. 



£r 



5* 

6- 
6- 
64 
64 





*!' '•■ 


8- ! 
8-^, 
9- 



l» o 



12-497 


1 

' 9-^ 


16-880 


lo- 


13-628 


ll- 


18-395 


12- 


14-765 


13- 


19-908 


14- 


15-897 


15- 


21-415 


15- 


17-034 


16- 


22-932 


16- 


24-447 


i 17- 


25-961 


t 17- 


27-471 


18- 


28-985 


: 18- 



® 2 



30-598 
32-208 
35-200 
38-456 
41-456 
44-640 
47-646 
59-588 
50-752 
63-470 
53-856 
67-344 
57-037 
71-258 



5 



19- 
19- 
20- 
21- 
22- 
23- 
24- 
25- 
26- 
27- 
28- 
29- 
30- 



u 


ii 


4 


60-142 


5 


75-233 


5 


78 208 


5 


82-984 


5 


86-771 


5 


90-571 


5 


94-308 


5 


98-122 


5 


101-897 


5 


105-700 


5 


109-446 


5 


113-221 


5 


116-997 



The above weights include the laps on the sheets for riveting and 
caulking. 

The weights of the rivets are not added ; the number per linear 
foot of pipe depends upon the distance they are placed apart, and 
their 5/^^ upon the diameter of the pipe. ; 

TABLE I 

Shoxcing the Capacity of Cisterns in Gallons. 
For each 10 Inches in Depth. 



Feet 
Diam. 



2 

2^ 

3 

3^ 

4 

4^ 



19-5 

30-6 

44-06 

59-97 

78-33 

9914 



Feet 
Diam. j 



5 

6 
6i 

7 
7^ 



122-40 
148-10 

176-25 
206-85 
239-88 
275-40 



Feet 
Diam 



8 

8^ 

9 

i n 
1 10 

; 11 



313 

353 

396 
461 
489 
592 



33 
72 

56 
40 
20 
40 



Feet 
Diam. 



12 


705- 


13 


827-4 


14 


959-6 


15 


1101-6 


20 


1958-4 


25 


3059-9 



* Weight of Copper, Lead, and Iron. 



276 



TABLE 

CorUaining the weight of a Square Foot of Copper and Lead in lbs. 

avoirdupois, from 3V ^^ \ <*^ inch in thickness, advancing by 3L. 



Thickness. 



32 

tV 

3 

32 

i 



i and ^L 



I 



3 2 



Copper. 


Lead. 

1 


1-45 


1-85 


2-90 


3-70 


435 


5-54 


5-80 


7 -39 i 


1-26 


9-24 \ 


%'n\ 


11-08 


10-16 


12-93 


11-61 


14-77 1 



Thickness. 



Copper. 



i and 



32 

1 

JL 

32 



JL 

32 
1 

T^ 

3 

32 



13-07 
14-52 
15-97 
17-41 

18 -87 
20-32 
21-77 
23-22 



Lead. 



16-62 
18-47 
20-31 
22 16 
24-00 
25-85 
27-70 
29*55 



TABLE 

Of the Weight of a Square Foot of Sheet Iron in lbs. avoirdupois, the 
thickness being the number on the wire gauge. — No. 1 is -^^ of an 
inch; No. 4. i; No. 11, i, (kc. 



No. on wire 
gauge, . 


1 


2 


3 


4 


5 


6 


7 


8 


Pounds 
avoirdu , 


12.5 


12 


11 


10 


9 


8 


7-5 


7 


No. on 
gau^ 

Pound 
avoi 


wire 


9 


10 


n 


12 


18 
4-31 


14 


15 
3-95 


16 


5 

rdu., 


6 


5-68 


5 


4-62 


4 


3 


No. on wire 
gauio;e, . 


17 


18 


19 


20 


21 


22 




Pounds 
avoirdu.. 


2-5 


2-18 


1-93 


1-62 


1-5 


1-37 




TABLE 

Of the Weight of a Square Foot of Boiler Plate Iron, from ^ to \ 

inch thick, in lbs. avoirdupois. 


i 
5 


^1 
7-5 


i 
10 


tV 
12-5 




5 17-5 20 


225 26 


27- 


5 30 32-5 


i 

36 


H 

87-5 


1 in, 
40 





276 



Pumps. 



TABLE 



SJiowing the Qaaiiiity of Water per Linear Foot in Piiwps, or 
Vertical Pipes of different Diameters. 



Diiimeter of 
pump in 
inches. 






H 
H 

3J 

4 

4i 

4i 
5 

5J 
5i 
5f 
G 

H 
6i 

6f 

7 
7i 



Nunnber of ! Number of 
gallons per jcubic feet per 
linear foot, i linear foot. 



•136 

•172 

•212 

•257 

•306 

•359 

•416 

•478 

•544 

•614 

•688 

•767 

•850 

•937 

1-028 

3-124 

1-224 

1-328 

1 -436 

1-549 

1-666 

1-787 

1-912 

2^042 



•0218 
•0-276 
•0340 
•0412 
•0490 
•0576 
•0668 
•0766 
•0872 
•0985 
•1104 
•1230 
•1363 
•1503 
•1649 
•1803 
•1963 
•2130 
•2304 
•2489 
•2672 
-2S66 
•3067- 
•3275 



Diameter Number of 

if jHimp g-alloiis per 

iii inches, jmear loot. 



8f 
9 

H 

10 

loi 

lOi 
lOf 

11 

Hi 
Hi 

iif 

12 

121 

13" 

131 

14 

15 

16 

18 



2-176 
2-314 
2-456 
2-603 
2-754 
2-909 
3-068 
3-232 
3-400 
3-572 
3-748 
3-929 
4^114 
4-303 
4-496 
4-694 
4-896 
5-312 
5746 
6196 
6-664 
7-650 
8-704 
11-016 



Number of 
cubic feet 
per linear foot. 



•3490 
•3712 
•3940 
•4175 
•4417 
•4666 
•4923 
•5184 
•5454 
•5730 
•6013 
•6302 
•6599 
•6902 
•7212 
•7529 
•7853 
•8521 
•9217 
•9939 
1-0689 
1-2271 
1-3962 
1-7670 



JSxampl-es illustrative of the Utility of the Table. 
1. Required the quaatity of water lifted by each stroke of the 
bucket of a 9|^-inch pump, the length of the stroke being 2^ feet. 

3 068 X 2 25 r= 6-903 gallons, each stroke. 
2 What length of stroke with a 6-inch pump will be necessary 
to discharge 44 gallons of water per minute, the number of strokes 
being 18 in the given time ? 

44 

— ~ To" ~ ^ ^^^^' ^^^^ length of stroke. 

3. What must be the diameter capable of raising 25 cubic feet of | 
water per minute, the length of the stroke being 2^ feet, and{ 
making 16 effective strokes per minute ? 1 



Screws. 



277 



25 



2-6 X 16 



•625, or lOf inches, nearly. 



Properties of Atmospheric Air. — It is by the oxygen of the 
atmosphere that combustion is supported. The common combusti- 
bles of nature are chiefly compounds of carbon and hydrogen, which, 
during combustion, combine with the oxygen of the atmosphere, and 
are converted into carbonic acid and watery vapor, different speoies 
of fuel requiring different quantities cf oxygen. Tlie quantity re- 
quired for the combustion of a pound of coal varies from two to three 
lbs. Sixty cubic feet of atmospheric air will produce 1 lb. of oxygen. 

The pressure or fluid properties of the atmosphere oppose bodies 
in passing through it, the opposing resistance increasing as the 
square of the velocity of the bod}', and the resistance per square 
foot in lbs. as its velocity in feet per second, multiplied into -002288. 
Thus, suppose a locomotive engine in a still atmosphere, at a velo- 
city of 25 miles per hour, presents a resisting frontage of 20 feet ; 
required the amount of opposing resistance at that velocity. 

25 miles per hour equal 36 '67 feet per second. 

Then 36*67^ x '002288 x 20 = 61-5 lbs., constant opposing force. 



TABLE 

Showing the Number of Threads to an Inch in V-thread Screws. 



Diam. in inches, 
No. of threads. 


20 18 


f 

16 


14 12 


11 


t 

in 


i 



1 

8 


H 

1 


7 


If 

6 




Diam. in inches, 
No. of threads. 


6 


If 
5 


If 
5 


n 

4-i 


2 
4| 


2i- 2i 
4 4 


2f 


3 

H 








Diam. in inches, 
No. of threads, 


3| 4 
3 3 


4i 


4i 4f 5 5J 5i 
21 2f 2| 2f 2| 


5f 


6 



Tlie depth of the threads should be half their pitch. The dia- 
meter of a screw, to work in the teeth of a wheel, should be such 
that the angle of the threads does not exceed 1 0°. 



24 



278 




Woods— 


-Gases. 






1 




TABLE 


Of the component parts of one Ey 


iglish pound avoirdupois of 1000 


grains of the following varieties of Wood. [Musiiet.] 




Water, 






Color and decree of saturation 


Description of Wood 


Carb. acid. 


Carbon. 


Ashes. 


ofthe alkaline principle. 


Oak, . . . 


5382-6 


1587-8 


29-6 


grey, sharply alkaline. 


j Ash, . . . 


5688-2 


1258-0 


53-8 


whitish blue, shrp. alk. 


1 Birch,. . . 


5650-2 


1224-4 


125-4 


brownish red, shrp. alk. 


: Norway Pine, 


5630-9 


1344-3 


24-8 


brown, not at all alk. 


Mahogaii}', . 


5147-0 


1784-4 


68-6 


grey, sharply alkaline. 


Sycamore, . 


5544-0 


1381-4 


74-6 


pure white, weakly alk. 


Holly,. . . 


5524-4 


1394-3 


81-3 


pure white, sharply alk. 


Scotch pine. 


5816-7 


1151-9 


31-4 


brown, perceptibly alk. 


Beech, . . 


5737-3 


1395-9 


66-8 


greyish white, shrp. alk. 


Elm, . . . 


5576-6 


1370-2 


53-2 


^rey, partially alkaline. 


Walnut, . . 


5496-5 


1446-4 


57-1 


5 pure white, light as 
( down, weakly alk. 


American Mapl 


e 5558-2 


1393-1 


53-7 


dark grey, sharply alk. 


Do. Black 
Beech, . 


- 5425-9 


1301-8 


72-3 


brown, sharply alkaline. 


Laburnum, . 


5196-4 


1721-0 


82-6 


white & grey, partly alk. 


Lignum Yitse, 


5083-0 


1880-0 


35-0 


2:rey, sharply alkaline. 


Sallow, . . 


5626-0 


1294-8 


79-2 


light gre}^ sharply aUc 


Chestnut, 


5341-3 


1629-6 


29-1 


grey, sharply alkaline. 




TAB 


LE 


1 
1 


Of Properti 


es of Gases. 


Atmo 


spheric air being the sta 


ndard of comparison, or 1000. 


Names. 


Specific 
gravity. 


Names. 


Specific 
giuvity. 


Hydriodic acic 


i gas, . 


4340 


Carbonic oxide gas, . 


972 


Chlorine " 




2500 


Carbaretted hy- 




Carbonic " 
Nitrous oxide 




1527 
1527 


drogen ^^ ^"^^ 
Prussic acid ^* 


972 
987 


Cyanogen 




1806 


Ammoniacal " 


590 


Oxygen 

i 

i 




1111 


Steam of water " 
Hydrogen *' 


623 

69 


1 









Screws — Atmosphere. 279 








TABLE 


i 


Of Change Wheels for Screw-cutti7ig ; the leading Screw being of -J 


inch pitch, or containing 2 threads in an inch. 


a Number of 


j s: 


i 




lil 




1^ 


teeth .a 


■3 b: 


Number of teeth in 


1 "^ 


Number of teeth in 


4) <X> 


















5 ^ j o I 


i« 


i 


If 1 If 


> 











^ 


1^'-^ 1 z: 5: 


! SS o 


'O 


= • 1 = ;^ 


OJ 


x: 




c ;, 











it 


!i i n 


tfllj 


i *c 




81 


?^ 


ll 


^ G ■ ' -G 1 C -S 




T.-^ 


— .5 1 -s 


c-fi 




-G 








^^1 


1 5 


1 


3 


^i 1 i^ 




S 
19 


5^ 








1 


80 


40 


H 


40 


00 


20 


60 


50 


95 


20 


100 


li 


80 


50 


H 


90 


85 


20 


90 


\H 


.80 


120 


20 


130 


H 


80 


60 


8f 


60 


70 


20 


75 


20 


60 


100 


20 


120 


If 


80 


10 


n 


90 


90 


20 


95, 


201 


40 


90 


20 


90 


2 


80 


90 


n 


40 


60 


20 


65| 


21 


80 


120 


20 


140 


2i 


80 


90 


10 


60 


75 


20 


80, 


22 


60 


110 


20 


120 


2i 


80 


100 


10^ 


50 


70 


20 


75; 


22^ 


80 


120 


20 


150 


2f 


80 


110 


11 


60 


55 


20 


120: 


22| 


80 


130 


20 


140 


3 


80 


120 


12 


90 


90 


20 


120 


23f 


40 


95 


20 


100 


H 


80 


180 


12f 


60 


85 


20 


90| 


24 


65 


120 


20 


130 


H 


80 


140 


13 


90 


90 


20 


130| 


25 


60 


100 


20 


150 


3| 


80 


150' 


131- 


60 


90 


20 


90 


251 


30 


85 


20 


90 


4 


40 ; 80: 


131- 


80 , 


100 


20 


110 


26 


70 


130 


20 


140 


4i 


40 ! 85' 


14 


90 i 


90 


20 1 


140| 


27 


40 


90 


20 


120 


4i 


40 1 90' 


14i 


60 1 


90 


20 


95! 


271 


40 


100 


20 


110 


4f i 40 1 95 


15 


90 


90 


20 


Io0| 


28 


75 


140 


20 


150 


5 


40 ; 100 


16 


60 1 


80 


20 


120' 


28* 


30 


90 


20 


95 


5^ 


40 


no; 


16i 


80 


100 


20 


130' 


30 


70 


140 


20 


150 


e> 


40 


120 


16i 


80 


110 


20 


120: 


32 


30 


80 


20 


120 


H 


40 


130 


17 


45 


85 


20 


90' 


33 


40 


110 


20 


120 


1 


40 140 


m 


80 


100 


20 


140! 


34 


30 


85 


20 


120 


n 


40 150 


18 


40 


60 


20 


120 


35 


60 


140 


20 


150 


8 


30 120 


18f 


80 


100 


20 


150 


36 


30 


90 


20 


120 


Temperature and T 


Vcigh 


t of the Atmosp) 


lere at various heights. 






! 




Water heavier than the 


Height. 






Temperatu 


re. 


air. 


Level of the sea, 






60° 




860 times. 


One mile above, 


, 




43 




1,083 " 


Two miles above, 






26 




1,363 " 


Three miles ab/ove, 






9 




1,716 " 


Four miles above, 






-8 




2,160 " 


Five miles above, 


' • 


-25 


2,719 " 1 



280 



Weight of Water in Pipes. 



TABLE 

Shoidng how to discover the Quantity and Weight of Water in Pipei 
of any given size. 



Di.imeter in 


! Quantity in cubic 


Quantity in imperial 


Weight in lbs. 


inches. 


inches. 


gallons. 


avoirdupois. 


^ 


14-14 


0-051 


0-51 


1 


! 56-00 


0-205 


2-05 


H 


127-23 


0-460 


4-60 


2 


226-19 


0-818 


8-18 


2i 


353-43 


1-278 


12-78 


3 


508 94 


1-841 


18-41 


34- 


692-72 


2-506 


25 06 


4 


904-78 


3-272 


32-72 


4i 


1145-11 


4-142 


41-42 


5 


1413-72 


5-113 


51-13 


H 


1710-60 


6-187 


61-87 


6 


2035-75 


7-363 


73-63 


6^ 


2389-18 


8-641 


86-41 


7 


2770-88 


10-022 


100-22 


n 


3180-86 


11-505 


115-05 


8 


3619-11 


13-090 


130-90 


H 


4085-64 


14-777 


147-77 


9 


4580-44 


16-567 


165-67 


n 


5103-52 


18-459 


184-59 


10 


5654-87 


20-453 


204-53 


lOi 


6-234-49 


22-550 


225-50 


11 


6842-39 


24748 


247-48 


lU 


7478-56 . 


27-049 


270-49 


12 


8143-01 


29-452 


294-52 



This table shows the quantity and weight of water contained in 
one fathom of length of pipes of different bores from 1 inch to 12 
inches in diameter, advancing by half inch. The weight of a cubic 
foot of water is taken at 1000 ounces avoirdupois, and the impe- 
rial gallon at 10 lbs. 



Multipliers used for ascertaining the quantity of Tallow^ Oakum.^ and 
Oil that can he contained in Tanks for use of Steam-vessels. 

Tallow, .... 59 lbs. in a cubic foot. 

Oakum, .... 11 lbs. in a cubic foot 

Oil, 6-23 galls, in a cubic foot 

Coal, 45 cubic feet to a ton. 





Specific Gravities 


AM) OTiiEii Properties of Bodies. 




281 


If 


Brass, cast, 

" sheet, 

Iron, cast, 

" bar. 
Steel, soft, 

" hard. 


Platinum, . . 
Pure gold, . . 
Mercury, . . . 
Lead, .... 
Pure silver, . . 
Bismuth, . . . 
Copper, cast, 
" wrought, 






1 

i! 

f 
f 

i 

a.- 

r 

II 

IS 

^« 

09 

a 

1 

1 


3 

CD 
CD 


f3 
o 

% 

i 

H 


• • 






t— to 

CO CO 

O ^ 


-:r ~q- -a- -q- 00 ^3" 

OO GO -T to W OO 

)-• 03 O Oi CO to 
05 OO O rf^ Oi hf^ 


OOOOcOOi— 'COCOCO 
O -J OO ►f^ CO oi to ox 
(--OOtO-JOxOOxO 
OOOCOi^^tOOOOO 


Specific gravity. 


OS to 


1 1 1 g 1 i 

05 O 


^ t-k to CO 
1 CO i4^ 00 Ci 1 o to 
1 CO -T -J ^ 1 h- 00 

05 OS CO to C5 O 


Melting- points in 
degrees oi Fah. 


CO to 

to -J 

CO 00 


1 >^ ^ ^ \ to 

1 CO 03 to 1 r-» 

CO -I OX O 


1 ss;| iil 1 1 

CO OS CO 


Contraction in 
parts of an inch 
per linear loot 
frqm the fluid to 
the average temp, 
in solid state. 


Ot to 


Oi to H-» 
1 00 ox -J to 00 
1 CO O 00 to o 

— ^ O -3- CO 1— ' 


oJ QO H- 1 1 1 I 

OC7.4S. 1 OO > 1 ' 

CO 1— ' C>X 1—1 


Ultimate cohesive 
strength of an iii. 
sq. prism in tons 


-:r 00 


1 1 ^ 1 o= 1 


Ol 1 1 to 00 1 ^ CO 


Scale of wire- 
drawing ductility. 


00 f^i' 


1 1 " 1 => 1 


CO 1 1 to -:t 1 i-« c;^ 


Scale of laminable 
ductility 


( to any 
\ degree 

{to any 
degree 
4-7 
( to any 
\ degree 
1-2 
1-6 


lnO j to to v-» 1 t-J 1 

GO ' <:5 ^i^ <Z> ^ oc ' 


Ratio of hardness. 


-0- Or 


1 1 ^^ 1 1 1 


t-i 1 1 to C5 1 CO 1 


Scale as conduc- 
tors of electricity. 


W 05 

1^ o 


1 1 S 1 S 1 


00 1 1 CO t-" 1 O CO 

CO ' ' -J ob ' o 00 


Ratio of power in 

the conduction of 

heat. 


Marble, average 
Granite, do. 
Purbeck stone, 
Portland do., . 
Bristol do , 
Millstone, . . 
Paving stone, . 
Craigleith do.. 
Grindstone, 
Chalk, British, 
Brick, . . . 
Coal, Scotch, . 
" I^ewcastle, 
" Staffordsh. 
" Cannel, . 


5^ 
p 
I 


i 

o 

•=3 

o 

> 
so 

H 

Jo 




»— ' •— ' — 1— ' lO to to 

to to to CO O ---T — 
CO >^ -J O O 00 4^ 
00 O O O O — CO 


lO to to lO fM, to to to 

ao^^*i.oxoxa>C5-3- 

05 )-• 00 O^ -T O Ox to 
toO'4i'>|i-Oi-'>— O 


Specific gravity. 




-J -3 -T 00 to -3^ CO 

•-T -J CO 1— C?l CO CO 


>^oxcnc;xai0505--T 
^TOCJxcoOtOCJxO 


Weight of n cubic 
foot in lbs. 


CO Ox CO t—" O 00 CO 
-J O --T 0\ O H-t CO 


C5cotoo505c;xCiO 

tOCOOxiOtOOiOOO 




to to to to t-i h-i 1-* 
CO CO --T ~J -J^ to 05 


Oi 4^ rfi. >fi. ff^ CO to CO 


Cubic feet in Eton. 




O O Oi 


O' Ox >fi' CO Oi CO 1 


Tons required to 
crush l><s in. cubes 


1 1 1 looox350-^l loiotoro 

Ox 1 


p 



24^ 



282 Specific Gravities and other Puoperties of Bodies. 



Properties of Woods. 











£ 


o 


' tc 




1 o 




■ y: 








y 


a 


u 


^ 



Englisli oak, : 934 

Riga do, . . . c 
Daiitzic do., .... 
American do., . . . 

Beech, 

Alder, 

Plane, 

Sycamore, .... 

Chestnut, 

Ash, 

Elm, 

Mahogany, Spanish, . 
" Honduras, 

Walnut, 

Teak, 

Poona, ...... 

African oak, . . . . 

Poplar, . . . . . 

Cedar, 

Riga fir, . . . . . 
Memel do., .... 
Scotch do., .... 
Christ, white deal, 
American white spruce. 
Yellow pine, . . . 
Pitch pine, .... 

Larch, 

Cork, 



872 

156 

072 

852 

800 

640 

604 

610 

845 

673 

800 

637 

671 

750 

340 

944 

383 

561 

753 

546 5 

528 

590 

551 

461 

660 

530 

240 



58 
54 
47 
42 
48 
46 
40 
38 
38 
52 
42 
50 
40 
42 
46 
40 
59 
34 
33 
47 
34 
33 
37 
34 
28 
41 
31 
15 



384- 
4U 
48 
53 
45 
48^ 
55 
59 
59 
43 
53 
45 
55 
53 
48* 
55 
38 
66 
68 
48 
66 
68 
60 
66 
80 
54* 
72 
149 



' O W 



g « « 



11880 

12888 

12780 

10253 

12225 

9540 

10935 

9630 

10656 

14130 

9720 

7560 

11475 

8800 

12915 

12350 

17200 

5928 

7420 

9540 

9540 

7110 

12346 

10296 

11853 

9796 

12240 



Comparative 



100 
93 

117 

114 
77 
63 
78 
59 
67 
89 
78 
73 
93 
49 

126 
99 

101 
44 
28 
98 

114 
55 

104 
72 
95 
73 
79 



100 
108 
107 
86 
103 



100 

125 

99 

64 

138 



801 101 

92 ! 108 

81! Ill 

89; 118 

119! 160 

82 1 86 

67 I 61 

96 1 99 

74| 111 

109! 94 

104 i 82 



144 
50 
62 

80 
80 
60 
104 
86 



138 

57 

106 

64 

56 

65 

104 

102 

99 I 103 

82 j 92 

103' 134 



Fusing Point of varioics Metals. 

The fusing points of the more refractory substances are only to 
be ascertained approximately, on account of the doubtful accuracy 
of the indications given by the pyrometers at very high tempera- 
tures. 

The pyrometer constructed of platinum is the most delicate, 
although the rate of its expansion must be uncertain as it ap- 
proaches its own fusing point. The following are considered to be 
the fusing points of metals: 



Specific Gravities and other Properties of Liquids. 



283 



Platinum, 
Wrought iron, 
Steel, . . . 
Gold, . . . 
Cast iron. 
Copper, 



Fahr. 
3080° 
2910 
2500 
2190 
2100 
1920 



Silver, 1830° 

Zinc 700 

Lead, 590 

Bismuth, 500 

Tin, 450 



A dull red heat is estimated as 1480° ; a bright red heat as 
1830°; and a white heat as 2370° to 2910^ Fah. 

Table of Properties of Liquids. 





>■ . 

CC o 


C3 — 




> . 


5'Z 
















bc = 


'oo 




fcXO 


"o^ . 


Names. 




tsi' 


Names. 








« C3 


<u d 




^ p 


3 a 




TO ^ 


^s 




OC '' 


^.s 


Acid, sulphuric, . 


1850 


18-5 


Oils, expressed: 






" nitric, . 


1271 


12-7 


linseed, . . . 


940 


9-4 i 


" muriatic, . 


1200 


12-0 


sweet almond, . 


932 


9-3 1 


" fluoric, . . 


1060 


10-6 


whale, . . . 


923 


9-2 ! 


" citric, . . 


1034 


10-3 


.hempseed, . . 


926 


9-3 1 


" acetic, . . 


1062 


10-6 


olive, .... 


915 


9*2 1 


Water from the 






Oils, essential : 




i 


Baltic, . . . 


1015 


10-2 


cinnamon, . . 


1043 


10-4 1 


Water from the 






lavender, . . 


894 


8-9 ! 


Dead Sea, . . 


1240 


12-4 


turpentine, . . 


870 


8-7 ! 


Water from the 






amber, . . . 


868 


8-7 ! 


Mediterranean, . 


1029 


10-3 


Alcohol, .... 


825 


8-2 ! 


Water, distilled, . 


1000 


10-0 


Ether, nitric, . . 


908 


9-1 ; 








Proof spirit, . . 


922 


9-2 ■ 






! 


Vinegar, . . . 


1009 


10-1 i 



Axle Grease. 

1. The popular axle grease of the celebrated Mr. Booth is made 
as follows : 

Dissolve \ lb. common soda in 1 gallon of Avater, add 3 lbs. of 
tallow and 6 lbs. of palm oil (or 10 lbs. of palm oil only). Heat 
them together to 200° or 210° Fah. ; mix, and keep the mixture 
constantly stirred till the composition is cooled down to 60° or 70°. 

2. Another and thinner composition is made with \ lb. of soda, 1 
gallon of water, 1 gallon of rape oil, and \ lb. of tallow, or palm i 
oil. " I 

3. The French compound, called Liard, is thus made : — Into 50 
parts of finest rape oil pnt \ part of caoutchouc, cut small. Apply 
heal, until it is nearly all dissolved. 

4. Mankettrick's lubricating compound consists of 4 lbs. of 
caoutchouc (dissolved in spirits of turpentine), 10 lbs. of common 



284 



Fusibility of Metals. 



'soda, 1 lb. of glue, 10 gallons of oil, and 10 gallons of water. Dis- 
j solve the soda and glue in the T\-ater by heat, then add the oil, and 

lastly the oaoutchouc, stirring them until perfectly inoorporated. 
5. The following is the ordinary kind of axle-grease in common 

use: — 1 part of fine black lead, ground perfectly smooth, "with 4 

parts of lar«i Some recipes add a little camphor. 





TABLE 








Of Fimhility of Metals, 








As given by M. Thenard. 






1. 


— Fusible helrnc a red heat. 

CENTIGRADE. 






Mercury, . . . 


— yy- 




Potassium, . . . 
Sodium, . . . 


-roS 


j Gav Lussac and 
1 I thenard. 




90 


' Do. do. 


Tin. 


210 


Xewton. 




Bismuth, . . . 


. 1 256 


Do. 




Lead, .... 


. 1 260 


Biot. 




Tellurium, . . . 


. 'a httle less fus. than lead 


Klaproth. 




Ai-senie, . . . 


undetermined 






Zinc, .... 


. . 370 


Brongniart. 




Antimony, . . 


. |» litUe below a red heat 






o 


—Infusible below a red heat. 




1 


PYROMETER OF WEDGWOOD. 






Silver, 


20^ 


Kennedy. 




Copper, .... 


•^' i 


AVedsrwood. 




Gold, 


32 


Do. 




Cobalt, 


\ a little less difficult l 
") to melt than iron f 






Iron, 


130 


Wedgwood. 




4< 


158 


Sir G. McKenzie. 




Mangranese, . . 


160 


Guy on. 




Nickel.. . . . 


160 


Richter. 




PalladiuTu, 
Molybdenum, 
Uranium, . . . 


Xearly infusible, and to 
! be obtained at a forge 
j* heat only in small 


- 
J 
1 


1 


Tungsten, . . . 


buttons. ' , 




■ 


Chromium, . . 


} 1 




■ 


Titanium, . . . 


^! 


:^ 


■ 


Cerium. . . . 


i Infusible at the forge 




■ 


Osmium, . . . 


1 furnace. Fusible at 




■ 


Iridium, . . . 


\- the oxy - hydrogen: 


f * 


■ 


Rhodium, . . . 


j blowpipe' 




■ 


^ Platinum, .... 


1 ; 




■ 


Colnmbium. . . . 


J> 




1 


1 



Discharge of Water. 



285 







TABLE 






Containing the Qu 
charged over a M 
the depths of th 


antities of Water, in cubic feet, that 
^cir per viiimie, for every inch in its b 
3 Water from the surface to the top 
exceed eighteen iuches. 


will be dis- 

readth, when 

edge of the 


wasteboard do not 




if 


ml 

e 


slii 
-IP 

1^ i^ 5 o 

a s 




I- = 

*- g 2 :i 

||£| 


1 ^.l^-H 
S S 


1 


0-403 


0-428 


10 


12-748 


13-535 


2 


1-140 


1-211 


11 


14-707 


15-632 


3 


2-095 


2-226 


12 


16-758 


17-805 


4 


3-225 


3-427 


13 


18-895 


20-076 


5 


4-50Y 


4-789 


14 


21-117 


22-437 


6 


5-925 


6-295 


15 


23-419 


24-883 


1 


'7-466 


7*933 


16 


25-800 


27-413 


8 


9-122 


9-692 


17 


28-258 


30-024 


9 


11*884 


10-564 


18 


30-786 


32-710 









TABLE 

Of the Composition of different Gunpowders. 



KINDS. 



i 



Ro^-al Mills at Waltham Ab- 
bey, England 

France, national establislim't . 
French, for sportsnnen, . . . 
French, for mining, .... 
United States of America, . . 

Prussia, 

Russia, 

Austria (musket), 

Spain, 

Sweden, 

Switzerland (a round powder) 

Chinese, 

Theoretical propor. (as above) 



75 

75 

78 

65 

75 

75 

73-78 

72 

76-47 

76 

76 

75 

75 



Charcoal. 



Sulphur. 



15 

12-5 

12 

15 

12-5 

13*5 

13-59 

17 

10-78 

15 

14 

14-4 

13-23 



10 

12-5 

10 

20 

12-5 

11-5 

12-63 

16 

12-75 

9 
10 

9-9 
11-77 



286 



Man and Horse Power. 



Alloys. 



1 Alloys having a Density greater than the 
! Mean of their Constituents. \ 



Gold and ziuc. 
Gold and tin. 
Gold and bismuth. 
Gold and antimony. 
Gold and cobalt. 
Silver and zinc. 
Silver and lead 
Silver and tin. 
Silver and bismuth. 
Silver and antimony. 
Copper and zinc. 
Copper and tin. 
Copper and palladium. 
Copper and bismuth. 
Lead and antimony. 
Platinum and molybdenum. 
Palladium and bismuth. 



Alloys having a Density less than the 
Mean of their Constituents. 



Gold and silver. 
Gold and iron. 
Gold and lead. 
Gold and copper. 
Gold and iridium. 
Gold and nickel. 
Silver and copper. 
:^.ilver and iron. 
Iron and bismutli. 
Iron and antimony. 
Iron and lead. 
Tin and lead. 
Tin and palladium. 
Tin and antimony, 
yickel and arsenic. 
Zinc and antimony. 



TABLE 

Shoicing the estimated Power of Man or Horse as applied to 

Machinery. 



Application of the Power. 



Lb.s. avr. at j Lbs. avr. at 

the rate of \ the rate of 

220 feet per \ one foot per 

minute. ! minute. 



A man is supposed to be capable of lifting \ 

or carrying \ 27 "2*73 

A man is supposed to be capable of turning 
the winch of a crane with a force equal 



to 



"When the united efforts of two men are 
applied to the winch of a crane, tlie han- 
dles being at right angles, each man 
exerts a force equal to 

A man is supposed to e.'vert a power in 
pumping equal to 

In ringing, a man exerts a force equal to . 

And in rowinsr, 

The power of a horse is equal to ... . 



28-637 



33-499 

17-335 
38-955 
40-955 
150 



6000 j- 



6300 



lor' 



7350. 

3814 
, 8570 
I 9010 
\33000 



I 
I 



Speed and Force of "Wind. 



287 



TABLE 

Of the Speed and Force of Wind, at different velocities. 



Velocity of the Wiiul in 



Miles per hour. Feet per second. 



1 

2 

3 

4 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

60 

80 

100 



1*47 

2-93 
4-40 

6-87 

7-33 

14-67 

22-00 

29-34 

36-67 

44-01 

51-34 

68*68 

66-01 

73*35 

88-02 

117-36 

146-70 



Force in lbs. avoir- 
dupois per square 
loot. 



*005 

•020 

•044 

'079 

•123 

•492 

M07 

1-968 

3-075 

4-429 

6-027 

7-873 

9-963 

12-300 

17715 

31-490 

49-200 



Common Appellations given to 
the Wind. 



Hardly perceptible. 
> Just perceptible. 

t Gentle, pleasant wind. 

A Pleasant, brisk gale. 

t Very brisk. 

t High windvS. 

j- Very high 



A storm or tempest 
A great storm. 
A hurricane. 
A violent hurricane, 
which wrenches and 
tears up trees, forces 
dwellings and minor 
buildings from their 
foundations,anddrives 
them before it. 



A'o/e.'-The following rule is used to find the force of wind acting perpendicularly upon 
a surface :— Multiply the surface in feet by the square of the velocity in feet, and the 
product by "002288. The result is the force in pounds avoirdupois. 

Table showi^tg the Height of the Boiling Pointy Fah., at different 

Heights of the Barometer, 



Barometer. 


Boiling Point. | 


Barometer. 


Boiling Point. 


Inches. 
31 

30^ 
30 
291 
20 


Degrees. 
213-57 
212-79 
212-00 
211-20 
210-38 


Inches. 

28^ 

28 

27-;^ 

27 


Degrees. 
209-55 
208-69 
207 '84 
206-96 


In a vacuum 
is more or less \ 


water boils at 98"* 
)erfcct. 


to 100°, according as the vaciuim 



288 



Powers of Various Spkcies of Fuel. 



TABLE 

Of the mze8 of NuU^ equal in strength to their Bolts. 



Diam. of Short diameter 


Diam. of 


Short diameter of 


Diam. of 


Short diameter 


bolt in in. 


ot nut in in. 


bolt in in. 




nut in inches. 


bolt in in. 


of nut in in. 


1 


1 


If 


2tV 


2i 


4t"^ 


* 


i 


H 


2|i 


2f 


4i 


^ 


i 


■ If 


2* 


2i 


m 


1 


h\ 


If 


3i 


2| 


5i 


f 


1>% 


1 n 


•31 


S 


5f 


i 


It", 


2 




3i 


51 


1 


If 


^ iii 


8i 


SA 


h\ 


n 


2 


1 2i 


4 


8i 


6f 


n 


2i 


: 21 


4i 


4 


H 



A'bZe.— The depth of the head should equal the diameter of the bolt; the depth of the 
nut should exceed it, in the proportion of 9 or 10 to 8. 



TABLE 

Showing the Power of various Species of Fuel. 



Species of Fuel. -f S.s 


Eftect in lbs. 
of water con- 
verted into 
steam of 220'^. 


Quantity to 
convert a cubic 
foot of water 
into low pres- 
sure steam. 

Quantity to 
convert a cubic 

foot of water 
into steam, al- 

lowinjr lo per 

cent, for loss. 


Caking coal, . . . 
Coke 


lbs. 
9800 
9000 


lbs. 

8-4 

n-n 

6-75 
5-13 
3 07 

2-8 


lbs. 

7-45 

8-1 

925 
12-2 
20-31 

22-5 


lbs. 

8'22 

9-00 


Splint coal, .... 
Oak wood, dry, . . 
Ordinary oak, . . . 
Peat compact, of or- ) 
dinary dryness, . ] 


7900 
6000 
3600 

3250 


JO'28 

13-6 

226 

250 



Ratio of the Hardnesses of Bodies. 



1^ 



TABLE 

Of the Ratios of the Sitccessivc Hardnesses of Bodies. 



Substances. 



Diamond from Ormiis, 
Pink Diamond, . . . 
Bluish Diamond, . . 
Yellowisli Diamond, . 
Cubic Diamond, . . 

li"by. 

Pale ruby, from Brazil, 
Deep blue sapphire, 
Do., paler, , . . 

Topaz, 

Whitish topaz, . . 
Ruby spinell, . . 
Bohemian topaz, . 
Emerald, .... 
Garnet, .... 

Agate, 

Onyx 





■&6 
is 


20 


3-7 


19 


3-4 


19 


3-3 


19 


3-3 


18 


3-2 


17 


4-2 


16 


0*0 


16 


3-8 


17 


3-8 


15 


4-2 


14 


3-5 


13 


3-4 


11 


2-8 


12 


2-8 


12 


4-4 


12 


2-6 I 


12 


2-6 



Substances. 



Sardon3'x, . . . 
Occidental amethyst. 
Crystal, .... 
Cornelian, . . . 
Green Jasper, . . 
Reddish yellow do. 
Schoerl, .... 
Tourmaline,^ . . 
Quartz, . . . . 

Opal, 

Chrysolite, . . . 
Zeolite, . ... 

Fluor, 

Calcareous spar, . 
Gypsum, .... 
Chalk, .... 



12 

11 

11 

11 

11 

9 

10 

10 

10 

10 

10 

8 

7 

6 



2-6 
2-7 
2-6 
2-7 
2-7 
2-6 
3-6 
3-0 
2-7 
2-6 
3-7 
2-1 
3-5 
2-7 
2-3 
2-7 



Ductilitij and Malleabilitij of Metals, 

Ductility is the property of being drawn out in length without 
breaking. This property is possessed in a pre-eminent degree by 
gold and silver, as also by many other metals, by glass in the liquid 
state, and by many semi-fluid resinous and gummy substances. 
The spider and the silkworm exhibit the finest natural exercise of 
ductility, upon the peculiar viscid secretions from which they spin 
their threads. AVhen a body can be readily extended in all direc- 
tions under the hammer it is said to be malleable ; and when into 
fillets, under the rolling press, it is said to be laminable. 

There appears, therefore, to be a real difference between duc- 
tility and malleability ; for the metals which draw into the finest 
wire are not those which afford the thinnest leaves under the 
hammei', or in the rolling press. Of this fact iron affords a good 
illustration. Among the metals permanent in the air seventeen are 
ductile and sixteen are brittle. But the most ductile cannot be 
wire-drawn or laminated to any considerable extent without being 
annealed from time to time during the progress of the extension, or 
ratlier tlie sliding of the particles alongside of each other, so as to 
loosen their lateral cohesion. 



25 



290 



DUCTFLITY OF MeTALS. 



i 

1 


TABLE 




Of the Rat 


to of the Ductility and Malleahility of Metals. 


Mefals <lucfile and 


Brittle metals in 
alphabetical order. 


Metals in the order 


Metals in the order 


jnalleable. In alpha- 
betical order. 


of their wire-drawing 
ductility. 


of '.heir laminable 
ductiiity 


Cadmium. 


Antimony. 


Gold. 


Gold.. 


Copper. 


Arsenic. 


Silver. 


Silver. 


Gold. 


Bismuth. 


Platinum. 


Copper. 


Iron. 


Cerium ? 


Iron. 


Tin. 


- Iridium. 


Chromium. 


Copper. 


Platinum. 


; Lead, 


Cobalt. 


Zinc. 


Lead. 


Magnesium, 


Columbium. 


Tin. 


Zinc. 


Mercury. 


Iridium. 


LeacL 


Iron. 


Kiekel. 


Maiio-anese. 


Xickel. 


XickeL 


.' Osmium. 


Molybdenum. 


Palladium? , 


Palladium ? 


' Palladium. 


Osmium. 


Cadmium? 


Cadmium ? 


Platinum- 


Rhodium. 


' 




Potassium. 


Tellurium. 






Silver. 


Titanium. 






Sodium. 


Tungsten, 






Tin. 


Uranium. 






■ Zine. 









Conducting Powers of Various Substances. 
The conducting power of wood is very low ; the softer woods 
being lower in this respect than those which are harder. Of metals, 
and some other substances, the following is the order, according to 
Despretz : 



Tin, 804 

Lead, 180 

Marble, 24 

Porcelain, 12 

Tile, 11 



Gold, 1000 

Silver, 973 

■ Copper, 898 

'.Platinum, S81 

"Lron,. 374 

Zinc,. 363 

Radiating Power of Various Substances. 
Bodies that have polished surfaces radiate heat less than those 
that are roughen-ed, and metallic surfaces less than those of more 
imperfect conductors. The following are the proportions of some 
of each, according to Leslie : 



Lamp-black, . . . .100 

■ Water, 100 

!■ Writing-paper, . , . 98 
Glass, ...... 90 

Tissue-paper, .... 88 

Ice, 85 



Rough lead, 45 

Mercury, 20 

Polished lead, .... 19 
Polished iron, . . . . 15 
Tin, silver, copper, and gold, 12 



Scale of Puoofs for Chaint Rigging. 



291 



Reflecting Powers of Various Substances. 
Heat is reflected from the surface on which its rays fall, in the 
same manner as light; the angle of reflection being opposite and 
equal to that of incidence. The metals are the strongest reflectors 
of heat, in the following order, according to Leslie : 



Lead, 60 

Tinfoil rubbed with mer., 10 

Glass, 10 

Glass, waxed or oiled, . 5 



Brass, J 00 

Silver, 90 

Tinfoil, 85 

Block-tin, 85 

Steel, 70 

Power of Various Substances to Transmit Heat. 
All bodies capable of transmitting heat are, raoj'e or less, trans- 
parent, though their powers of transmitting heat and light are not 
in the same relative proportions; as the following list of the rela- 
tive powers of equal masses, determined by Melloni, will show : 



Air, 100 

Rock salt, transparent, . 92 

Flint-glass, 67 

Bisulphuret of Carbon, . 63 
Calcareous spar, transparent, 62 

Rock-crystal, 62 

Topaz, brown, .... 57 

Crown-glass, ..... 49 

Oil of turpentine, ... 81 



Rape-seed XDil, . . 
Tourmaline, green, 
Sulphuric Ether, 
Gypsum, . . . . 
Sulphuric Acid, . . 
Nitric Acid, . , . 
Alcohol, . . . . 
Alum, in crystals, . 
Water, 



2 
7 
21 
20 
17 
15 
15 
12 
11 



TABLE 

Showing the Scale of Proofs for Chahi Pigging close-linked, d:c. ; 
the txtreme Lengtli of Links not to exceed five dictyneters of their 
size in Iron. 



Dtam. of 
Links. 



Inches. 

If 

If 
li 

IJr 
1 



Testing 
Weight. 


Max. 

Strain. 


Tons. 


Tons. 


31f 


75 


27 


64 


22f 


54 


18| 


45 


15i 


37 


12 


30 


m 


26 


H 


23 


n 


20 


6f 


17 



Minimum 
Strain. 



Tons. Cwt 
68 

58 

49 

41 

34 

28 

25 

22 

20 

16 



Diam. of 
Links. 


Test. 
Wght. 


Inches. 


Tons 


W 


5| 


f 


4f 





3f 


\ 


3 


1 
i7 


01. 

-4 


i 


If 




H 


3 2 


i 


i 


i 


r? 


1 n 

IJ4 



Maximum 
Strain. 



Tons. 
14 

12 

10 

7, 
6 
4 
3 
none broken. 

14 

ItV, 



Minimum 
Strain. 



Tons. Cwt. 
13 10 

10 15 

8|- nearly. 

6 18 

5 2 

3 

2 14 

none broken. 

1 14 

19 



292 Masonry. 



MASONKY. 

Of the different kinds of Masonry. 

Masonry, in the general acceptation of the terni, is the art of 
cutting or squaring stones, to be applied to the purposes of build 
ing; or, in a more limited sense, it is the art of joining stones toge 
ther with mortar, or otherwise. 

The ancients enumerate seven different methods in which they 
arranged the stones of their buildings. Yitruvius tlius classes 
them: three of hewn or squared stones, threw of unhewn, and one 
a mixture of both methods. 

1. li'et masonry. This is represented in fig. 83, within the area 
D EFG, where the stones are squared and 
placed upon one of the angles, their joints 
thus forming a net-like appearance. This 
method, though very neat, is wanting in 
firmness and strength; for the oblique 
position of the stones, in regard to each 
other, gives them a tendency to separate 
rather than to form a compact assemblage 
of parts that unite in supporting each 

other. Whenever this form of masonry is employed, it is conse- 
quently necessary to keep the work together by a border of stones, 
having some other arrangement, one that is not only capable of 
supporting itself, but of overcoming the resistance of the net-like 
form. This is shown in the same figure at ABC; and where the 
network is merely a casing of stone to the brickwork of a wall, it 
w^ill be found to answer tolerably well, and looks ver}' neat. 

2. JBoiind masonry is that represented in tig. 2, and is remark- 
ably strong. The perpendicular joints in each course fall directly 
in the middle of the stones composing the course below and above 
it; and while it has ever}' requisite of solidity, the joints have, at 
the same time, a regular and pleasing appearance. 





/^^ 


// 


/ / /JE/ / 


'C / 




4 

/ / 






/ 



Fig. 2. Fig. 3. 

3. Greek masonry is that represented in fig. 3, wdiere every 
alternate stone, as shown at AD, EF, and B C, is made of the 
whole thickness of the wall, and serves to bind together the stones 



Masonry. 



293 



Fig. 4. 



which compose the external and internal faces of the building ; and 
this may be called double binding, as from the perpendicular joints 
being somewhat similarly situated to that in bound masonry, it has 
also an additional binding, by extending to the courses above and 
below it thus forming a compact and durable wall, which resists 
every effort to separate in any direction. 

4. Masonry by equal courses. This method of uniting stones is 
shown in iig, 4, and only differs from the bound masonry in its 
being composed of unhewn stones, or rather 
in being formed of stones that are not so 
accurately cut, nor the edges so perfectly 
squared ; it being only necessary that the 
external face should be level, and the hori- 
zontal joints at equal distances from each 
other, care being taken at the same time 
that the perpendiculars are so situated as to 
bind the courses above and below them. 

5. Masonry by unequal courses. This is represented in fig. 5, and 
is, like the last, formed of unhewn stones, without any regularity 
as to their size, it being sufficient that 
each course is made to bind with the 
preceding, and the only regularity 
observed is in the joining which sepa- 
rates each course, the courses them- 
selves being of unequal thickness, as 
shown at A B C D. 

6. Maso7iry filled up in the middle, 
as shown in fig. 6, is formed of un- *'"■ ^' 

hewn stones of unequal courses, and the middle, as at D, is filled 
up with stones thrown in at random among the mortar. 



^B 



^^^^ ^\ 


1 1 1 


x> 


1 1 1 


I- ( 1 


1 1 1 


1 1 1 


J 


1 1 1 


■^ 




Fig. 6. 




Fig. 7. 



7. Compound masonry is, as its name imports, a mixture of the 
other kinds. It is represented in fig. 7, where the external course 
AB is formed of bound masonry, and the corresponding internal 
course is at some distance from it, but held to the former by moans 
of iron cramps, as shown at «, b, c, d, e, f, the space between being 
filled in with small stones or flints thrown into the mortar. 



25* 



294 Masonry. 



The Methods of Joining Stone. 

As the strength and durability of masonry depend as much on 
the method employed, and the care taken in making all the joints 
to correspond accurately with each other, as in the quality of the 
material employed, some remarks will be required in explanation 
of the methods of joining stone. We shall, therefore, enumerate 
the several means adopted by workmen, and, where necessary, 
notice the purposes to which each method is best adapted, giving 
some cautions to secure success in practice, and to save the work- 
man unnecessary labor and trouble. 

The joints in masonry are either secured by the means of mortar, 
cement, or plaster of Paris, or the courses are held together by 
cramps, joggles, mortice, and tenoning, or dovetailing. 

1. Joining by mortar, or cements. It is absolutel}^ necessary 
that the joints should be perfectly smooth, and touch in every 
part ; and the stones must be so square as to bed well on each 
other, that is to say, they must not have such irregular faces as to 
roll, or, in technical terms, be winding to each other. The greatest 
care must be taken by the workman to have his mortar of a proper 
consistence — not too thin, as in drying it would shrink from the 
work, nor too thick, for that would prevent the stones from bedding 
pro'perly. The best way in irregular masonry, or in that composed 
of small stones thrown, as it were, between the regular work, as in 
compound masonry, is to saturate fresh lime with water, and^ 
while hot, to pour it on the work, which hardens and consolidates 
the Avhole into one solid mass. This method is much used in join- 
ing soft stones and brickwork, and is calculated to promote the 
strength and solidity of the work. 

2. Joining by cramps. Cramping is performed by inserting into 
the two pieces of stone, which are to be bound together, a piece of 
iron or some other metal, the ends of which, bent at right angles, 
are inserted in a cavity cut in each stone, the cavities being so 
large as to admit the iron easily; melted lead is then poured in to 
fill the vacant space, and, when cold, a chisel is driven into it, so 
that it may press close to the work ; for all metals expand by 
fusion, and obstacles may prevent them from contracting in cooling. 
Cramps composed of copper are, in many cases, very preferable to 
those made of iron, for they are less likely to oxidize, or rust, or to 
be affected by the lime or mortar. It would be of advantage to 
coat the cramps, if made of iron, with some substance that would 
defend them from the effects of damp. We may here remark, that 
the channel made to receive the cramp should be dovetailed, to 
prevent the lead from coming out, which it is otherwise apt to do, 
in the course of time. The only objection to the use of copper 
cramps, in preference to iron, is their expense, which in large 
public works is not of any importance, and, for common purposes, 
iron answers very well ; but the more malleable or tough the iron 



Masonry. 295 

the better it is, as it is moi'e calculated to resist the different tem- 
peratures to which the work may be exposed. 

3. Joining by joggles. The method of securing the joints of 
masonry by means of joggles is chief!}' adopted for securing the 
joints of columns or pillars; and consists in sinking a cavity in the 
two pieces in such a manner as to make them correspond with each 
other, and inserting in that cavity a piece of metal, stone, or even 
wood, so that any lateral thrust may not be able to separate 
them. This method may, with very great advantage, be applied in 
the construction of domes, and works^of the same "nature, where it 
is necessary to avoid the lateral thrust as much as possible. 

We may here take the opportunity of mentioning a plan proposed 
by Dr. Hutton, in his edition of Oznamare's Mathematical Recrea- 
tions, for taking away the lateral thrust of domes and cupolas. The 
following is the problem proposed, and the solution given : ' 

^^ How to construct a hemhpherical arch, or what the architects call 
an arc en cul-de-four, which shall have no thrust on its piers. 

" Let A B, fig. 8, be two contiguous voussoirs, which we will sup- 
pose to be three feet in length, and eighteen inches in breadth. 
Cut out on the contig- 
uous sides two cavities, .'^^^^^^^ 

in the form of a dove- /^^""^ — 7"^^" 

tail, four inches in 7\/ -A- [^^^^^^^ 

depth, with an aper- ^---Z_ ^-^^-"^^ — J 

ture of the same ex- """ — 

tent, a, 6, five or six Fig. 8. 

inches in length, and as much in breadth. This cavity will serve 
to receive a double key of cast-iron, as shown in fig. 9, or of 
common forged-iron, which is still more secure, as it 
is not so brittle. These two voussoirs will thus be 
connected together in such a manner that they can- 
not be separated without breaking the dovetail at 
the re-entering angle; but, as each of its dimensions ^^s- ^^ 

in this place will be four inches, it will be easily seen that an 
immense force would be required to produce that effect; for we 
are taught, by well-known experiments on the strength of iron, that 
it requires a force of four thousand five hundred pounds to break a 
bar of forged iron an inch square, by the arm of a lever of six 
inches ; consequently, two hundred and eighty-eight thousand 
pounds would be necessary to break a bar of sixteen square inclies, 
like that in question. Hence there is reason to conclude, that these 
voussoirs will be connected together by a force of two hundred and 
eighty-eight thousand pounds; and as they will never experience 
an effort to disjoin them nearly so great, as might easily be proved 
by calculation, it follows that they may be considered as one 
piece." 

They might be still further strengthened in a very considerable 
degree, for the height of these dovetails might be maclo double, and 





296 



Masonry. 





Fig. 11. 



a cayity might be cut in the middle of the bed of the upper vous- 
soir, fit to receive it entirely: the dovetail could not then be I 
broken without breaking the upper voussoir also; but it may be | 
easily seen that, to produce this effect, an immense force would be i 
required. 

The second method proposed by Dr. Hutton is more properly by ! 

the aid of joggles. Let I 
A andB, fig. 10, be two 
contiguous voussoirs, 
and C, fig. 11, the in- 
verted voussoir of the 
next course, which 
^^°' ^0- ought to cover the 

joint between A and B. Each of the voussoirs A and B being 
divided into two parts, as a 6 and cd; then 
if at ab and c d we sink an hemispherical 
cavity, in wiiich to introduce a globe of 
very hard marble, and in the upper voussoir, 
fig. 10, we sink similar cavities, 6c; this, 
when laid on he, fig. 11, will form a secure 
joint w^ithout any lateral thrust ; and the two courses cannot be 
separated without a force adequate to either break the solid stone, 
or disunite the marble globe ; a force almost inconceivable, or at 
least one far superior to that produced by the arch ; the whole 
dome, or cupola, is, in fact, one solid mass, and can exert no lateral 
thrust upon the walls on which it is raised. Marble globes are 
recommended, because iron is liable to rust; but, if the joggles 
were made of iron, and covered with pitch before they were placed 
in the cavities, there would be little to fear from rust ; and particu- 
larly as the iron is inclosed in the substance of the stone, and quite 
excluded from the action of atmospheric causes. r | 

Little need be said in this place as to morticing and^t^noning, or 
dovetailing, except that they differ slightly from the same opera- 
tions in joiners' work; for, as cement is used in the joining, they 
need not be so accurately cut, and are made shorter and thicker 
than those formed by the joiner, it being sufficient that the j^arts of 
each piece to be joined enter into each other at most five or six 
inches, even in large masses of stone. In small pieces, an inch or 
an inch and a half is sufficient; for, if the tenon or dovetail be too 
long, it will decrease the solidity of the joint. For greater security, 
a small channel is frequently cut in the shoulder of the joint, and 
melted lead is poured into it, which, filling up the space round the 
tenon or dovetail, makes the joint more secure, and the work firm 
^nd solid. 

In laying some sorts of stones, it is desirable, as far as possible, to 
place tliem in the same direction as they had when in the quarry, 
or, as it is termed by workmen, bedways of the stone ; for, if laid 
in other directions, they are liable to peel and split by the action 
of the atmosphere. 



Bricklaying. 297 



BEICKLAYING. 

Foundations. 

Tlie best soils for building upon are gravel, chalk, and stone 
rock. 

Those most to be guarded against are sands, bog earth, clays, 
and made earth (no matter how hard). Where these occur, avoid 
piling (except in water works) ; plank the foundations through the 
centre of the walls, place long tassels in the piers, la}- in chain 
bond, let the plates be stout, and in one piece, the whole length of 
each wall ; all that is required is to so bind the building that it 
may settle altogether, and not partiall}'. 

In doubtful foundations, it is advisable to have a trench dug out 
to the depth of 2 feet to 3 feet below the footings of the brick- 
work, and about twice the width of the footings, Avhich is to be 
filled up with concrete^ composed of stone lime ground and ballast, 
or coarse gravel, to be mixed with water, in the Y)roportion of one 
of lime to five or six of gravel ; immediately that it is made up it 
must be shot into the trench from a stage, 6 feet above, which will 
cause it to fall in a solid mass; and in a fcAV hours afterwards it 
Avill be as firm as a rock. 

It is strongly recommended to have good plates; whatever may 
be slighted in other parts these should not be neglected — they are 
the soul and support of a building, and cannot, if put in too small, 
be taken out and re[)laced, as other timbers may ; the difference 
in large houses will rarely amount to twenty-five dollars. 

Bond the work — English bond — using all whole bricks, a course 
of stretchers and headers alternately. 

Particular care must be taken that all the internal joints of brick- 
work are well flushed up with mortar; too frequently the work- 
men are apt to neglect doing it ; the consequence is, that all the 
interior joints are hollow, and allow the damp to penetrate to the 
inside, no matter how thick tlie wall may be. Another serious 
defect in brickwork is in not properly bonding the facing to the 
backing, particularly if the facing be malms or bricks, which cost 
an extra price ; generally the headers are only bats or half bricks, 
instead of being a stretcher or a whole brick to bond in with the 
brickwork at the back; there ought to be at least one stretcher in 
every 3 feet to each course, if there be not the wall will split or 
divide into two thicknesses. 

In building arches of a large span, it is advisable to build them 
in half brick rims, with vertical or radiating bond every 3 or 4 feet 
in the girt; if this latter precaution be not adopted, the conse- 
quence will be, that when tlie centre is struck, the i-ims will divide! 
and Aveaken the arch, and perhaps cause a total failure. 

In selecting bricks, clap them together — if they ling well, and, 



.£t8 






Bricklaying. 








when broken, sliow that they are burnt through, they will answer 


the parpose. A hard clamp burnt gray stock is all that is wanted 


for strength ; for water-works and foundations use clinker burnt 


marl stocks. Avoid samnel or place bricks, and chuffy stocks, and 


generally prefer hand tempering to pugging the clay. 


In mixing of mortar, it is advisable to see that the laborer pro- 


perly turns" up the mortar, and that the lime is thoroughly incor- 


porated with the sand throughout ; avoid ushig too much water, as 


it drowns the linie and weakens it; in large works it is best to mix 


the lime and sand in a mill — cement must be mixed in small quan- 


tities. 


TABLE 


SJioioing the Quanlitj/ of Earth to he removed, the Nitmher of Bricks 


and Gallons hi one foot in depth or length. 




-| Brick rim. 


1 Brick rim. 




Diani. 
in the 
clear. 








Feet cube Number of bricks. 


Feet cube 

of 
digging. 


Number of bricks. 


Contents 

in 
gallons. 


ft. in. 


of 

digging. laid dry. 


in mortar 


laid dry. 


in mortar 




9 


1 8 


23 


19 


4 


60 


50 


2f 


1 


2 4 


28 


23 


4 9 


70 


58 


5 


1 3 


3 1 


33 


27 


5 9 


80 


66 


7i 


1 6 


4 


38 


31 


7 1 


90 


74 


11 


1 9 


4 9 


43 


35 


8 3 


102 


82 


15 


2 


5 9 


48 


41 


9 6 


112 


92 


19f 


2 3 


7 1 


53 


44 


11 


122 


100 


25 


2 6 


8 3 


58 


48 


12 6 


132 


108 


30^ 


3 


11 


68 


57 


15 9 


154 


126 


44 


3 6 


14 2 


79 


65 


19 6 


174 


142 


60 


4 


17 7 


89 


73 


23 8 


194 


159 


78 


4 6 


21 6 


100 


82 


28 3 


214 


176 


100 


5 


26 


110 


90 


S3 2 


234 


192 


122 


5 6 


30 7 


120 


98 


38 5 


254 


209 


149 


6 


85 8 


130 


107 . 


44 2 


276 


226 


176 


6 6 


41 3 


140 


115 


50 3 


296 


242 


206 


n 


47 2 


150 


123 


56 7 


316 


260 


239 


7 6 


53 5 


160 


131 


63 6 


336 


276 


275 


8 


60 1 


170 


140 ! 


70 9 


358 


292 


313 


8 6 


67 2 


180 


148 


78 5 


378 


308 


354 


9 


74 7 


191 


156 


86 6 


398 


326 


396 


10 


90 8 


212 


174 


103 9 


438 


360 


489 





Brick LAYING. 299 



In the measurement of brickwork no allowance is to be made in 
quantity for small or difficult works. 

Flues to be measured solid. 

Timbers inserted in walls not to be deducted. 

Two inches to be allowed for bedding plates, where no brick- 
work is over them. 

All cuttings to be measured superficially, excepting to bird's mouths 
and squint quoins, Avliich are to be run. 

The net quantity of brickwork being found, it is to be reduced 
to the standard thickness of a brick and a hfilf, and brought into 
statute rods of 5^ yards square, or 272 superficial. 

Ovens, coppers, and solid walls, of irregular thickness, to be 
cubed and brought into the standard thickness, by multipl3'ing by 
8 (the number of 1^ inches in a foot), and dividing by 9 (the num- 
ber of 1| inches in a brick and a half, or loj inches, the standard 
thickness). 

Facings of all descriptions to be measured and. charged extra, per 
foot superficial. 

272 feet superficial is a rod of brickwork, 1-^ brick, or 13-^ inches 
thick, the standard thickness, to which all brickwork, of whatever 
thickness, is reduced. 

306 cubic feet, or 11 J cubic yards, equal to 1 rod of reduced 
brickwork. 

4352 stock bricks to 1 rod reduced, 4 courses 1 foot high. 

4533 ditto, if the 4 courses measure 11^ inches high. 

These calculations are without allowing an}^ waste, which is 
more than amply compensated in dwelling-houses, by not deducting 
flues and bond timber ; in such work, 4300 stocks, or 4500 place, are 
sufficient. 

5371 bricks laid dry to 1 rod. 

4900 ditto in Avells and circular cesspools. 

A rod of brickwork contains 235 feet cube of bricks, and 71 feet 
of mortar (4 courses to a foot) ; which will weigh, upon an average 
calculation, 15 tons. 

A rod of brickwork requires 1t^ cubic yard of chalk lime, and 3 
single loads or yards of drift ; or 1 cubic yard of stone lime, and 
3^ single loads or yards of sand; or 36 bushels of cement, and 36 of 
sharp sand. 

16 bricks to a foot of reduced brickwork. 

7 ditto to a foot super of facing. 

10 ditto to a foot super of gauged arches. 

30 bricks on edge, and 45 bricks flat, to 1 3'ard of brick- 
nogging. 

36 stocks laid flat, and 52 ditto on edge, to 1 yard of paving. 

36 pavinff bricks laid flat, and 82 ditto on edge, ditto. 

A load of mortar, 27 feet cube, requires 9 bushels of lime and 1 
yard of sand. A hod contains 20 bricks. 

Lime and sand loses one third of its bulk when made into mortar 
— likewise cement and sand. 



BOO 



Plastering. 



The proportion of raortar, or cement, when made up, to the lime, 
or cement and sand before made up, is as 2 to 3. 

Lime, or cement and sand, to make mortar, require as much 
water as is equal to one third of their bulk, or about 54^ barrels for 
a rod of brickwork built with mortar. 



P L A S T E E I N G . 



Inch yards. 

• M • 

• H ■ 

• H . 
. 4i . 



X inch yards. 

. u . 



41 



% inch yds. 

. u 

. 9 



Thickness of Compo. 

1 bushel of cement will cover . 

1 do. and I of sand do. 

1 do. and 2 do. do. 

1 do. and 3 do. do. 

(f inch is the usual thickness.) 

1 cwt. of mastic and 1 gallon of oil . . . 1|- . . . 2J 

1 cubic yard of clialk lime, 2 yards of road drift or sand, and 3 
bushels of hair, will cover 75 yards of render and net on brick, and 
70 yards on lath, or 65 yards plaster and render 2 coats and set on 
brick, and 60 yards on lath ; floated work will require about the 
same as 2 coats and set. 

1 bundle of laths and 500 nails Avill cover about 4^ yards. 

Mortar. 

1 hundred of lime contains 25 striked bushels, or 100 pecks. It 
is a measure 3 feet square, and 3 feet 1 inch deep. 1 chaldron of 
lime is equivalent to 57*765 cubic feet, or rather more than 2 hun- 
dred. 

18 heaped bushels, 22 striked bushels, or 1 yard cube, a single 
load of sand, mortar, (fee. 

1 double load is equal to 36 heaped bushel.'^. 

1 hod of mortar is equal to 1134 cubic inches, or 8 duodecimal 
inches, or 9 x 9, and 14 inches long. 

2 hods of mortar make a bushel nearly. 

Cement. 

1 barrel of cement is 5 bushels, and weighs 3 cwt. 1 rod of 
brickwork, in cement, requires 36 bushels of cement and 36 bushels 
of sand. 

1 yard, or 9 feet superficial of 14 inches, or 1| brickwork, in 
cement, requires about 2^^ bushels. 

1 yard superficial of pointing to brickwork, in cement, requires 
about one eighth of a bushel. 

1 yard square of plastering, in cement, requires three fourths of a 
bushel. 



Plastering. 301 



Carpentry mid Plastering are measured by the square foot or 
yard ; or, iu moulded and ornamental work, by the linear foot. In 
extensive work the square of 100 feet is also used. 

Paving is measured by the square yard. 

Digging, etc. 

23^ cubic feet of sand, 17 J ditto clay, 18 ditto earth, 13 ditto 
chalk, equal to a ton. 

A cubic yard of earth, before digging, Avill occupy about 14- 
cubic yard when dug. 

27 cubic feet, or 1 cubic yard, contains 21 striked bushels, which 
is considered a single load, and double these quantities a double 
load. 

18 cubic feet of night soil, 1 ton. 

2|- tons of ditto is the quantity a cart contains ; 6 feet long, 3 
feet 3 inches wide, by 2 feet 4 inches deep, or 45 feet cube. 

Coarse Stuff. 

Coarse stuff, or lime and hair, as it is sometimes called, is pre- 
pared in the same way as common mortar, with the addition of 
hair procured from the tanner, which must be well mixed with the 
mortar by means of a three-pronged rake, until the hair is equally 
distributed throughout the composition. The mortar should be 
first formed, and when the lime and sand have been thoroughly 
mixed, the hair should be added by degrees, and the whole so 
tlioroughly united that the hair shall appear to be equally distri- 
buted throughout. 

Fine Stuff. 

This is made by slaking lime with a small portion of Avater, after 
which so much water is added as to give it the consistence of 
cream. It is then allowed to settle for some time, and the super- 
fluous water is poured off*, and the sediment is suff'ered to remain 
till evaporation reduces it to a proper thickness for use. For some 
kinds of work it is necessary to add a small portion of hair. 

Stucco for Inside of Walls. 

Tliis stucco consists of fine stuff already described, and a portion 
of fine washed sand, in the proportion of one of sand to three of 
fine stuff. Those parts of interior walls are finished witli this 
stucco which are intended to be painted. In using this material, 
great care must be taken that the surface be perfectly level, and to 
secure this it must be well worked with a floating tool or wooden 
trowel. This is done by sprinkling a little water occasionally on 
the stucco, and rubbing it in a circular direction with the float, till 
the surface has attained a high gloss. The durability of the Avork 
very much depends upon the care witli which tliis process is done, 
for if it be not thoroughly worked it is apt to crack. 

__ 



302 Plastering. 

Gauge Stuff, 

This is cliieily used for mouldings and cornices which are run or 
formed with a wooden mould. It consists of about one fifth of 
plaster of Paris, mixed gradually with four fifths of fine stuff. 
When the work is required to set very expeditiously, the proportion 
of plaster of Paris is increased. It is often necessary that the 
plaster to be used should have the property of setting immediately 
it is laid on, and in all such cases gauge stuff is used, and conse- 
quently it is extensively employed for cementing ornaments to 
walls or ceilings, as well as for casting the ornaments themselves. 

Higgind Stucco. 

To fifteen pounds of the best stone lime add fourteen pounds of 
bone ashes, finely powdered, and about ninety-five pounds of clean, 
washed sand, quite dry, either coarse or fine, according to the 
nature of the w^ork in hand. These ingredients must be intimately 
mixed, and kept from the air till wanted. When required for use, 
it must be mixed up into a proper consistence for working with 
lime water, and used as speedily as possible. 

Parker'' 8 Cement. 
This cement, which is perhaps the best of all others for stucco, 
as it is not subject to crack or flake off, is now very commonly 
used, and is formed by burning argiltaceous clay in the same 
manner that lime is made ; it is then reduced to powder, by the 
process described in a previous part of this work. The cement, as 
used by the plasterer, is sometimes employed alone, and sometimes 
it is mixed with sharp sand ; and it has then the appearance, and 
almost the strength, of stone. As it is impervious to water, it is 
very proper for lining tanks and cisterns. 

Hamelein^s Cement. 

This cement consists of earthy and other substances insoluble in 
water, or nearly so ; and these may be either those which are in 
their natural state, or have been manufactured, such as earthen- 
ware and china ; those being always preferred w^hich are least 
soluble in water, and have the least color. When these are pulve- 
rized, some oxide of lead is added, such as litharge, gray oxide, or 
minium, reduced to a fine powder; and to the compound is added 
a quantity of pulverized glass or flint stones, the whole being 
thoroughly mixed and made into a proper consistence with some 
vegetable oil, as that of linseed. This makes a durable stucco or 
plaster, that is impervious to wet, and has the appearance of stone. 

The proportion of the several ingredients is as follows: — to ever}^ 
five hundred and sixty pounds of earth, or earths, such as pit sand, 
river sand, rock sand, pulverized earthenware or porcelain, add 
forty pounds of litharge, two pounds of pulverized glass or flint, 
one pound of minium, and two pounds of gray oxide of lead. Mix 



Plastering. 303 



the whole together, and sift it throiigli sieves of different degrees 
of fineness, according to the purposes to which the cement is to be 
applied. 

The following is the method of using it: — To every thirty pounds' 
weight of the cement in powder add about one quart of oil, either 
linseed, walnut, or some other vegetable oil, and mix it in the same 
manner as any other mortar, pressing it gently together, either by 
treading on it, or whh the trowel; it has then the appearance of 
moistened sand. Care must also be taken that no more is njixed at 
one time than is required for use, as it soon hardens into a solid 
mass. Before the cement is applied, the face of the wall to be plas- 
tered should be brushed over with oil, particularly if it be applied 
to brick, or any other substance that quickly imbibes the oil; if to 
wood, lead, or any substance of a similar nature, less oil may be 
used. 

Maltha, or Greek Mastic. ^ 

This is made by mixing lime and sand in the manner of mortar, 
and making it into a proper consistency with milk or size, instead 
of water. 

Plaster in 'hnitatiou of Marble. 

This species of work is exquisitely beautiful when done with 
taste and judgment, and is so like marble to the touch, as well as 
appearance, that it is scarcely possible to distinguish the one from 
the other. We shall endeavor to explain its composition, and the 
manner in Avhich it is applied ; but so much depends upon the 
workman's execution, that it is impossible for any one to succeed 
in an attempt to work with it without some practical experience. 

Procure some of the purest g3'psum, and calcine it until the large 
masses have lost the brilliant sparkling appearance by Avhich they 
are characterized, and the whole mass appears uniformly opaque. 
This calcined gypsum is reduced to powder, and passed through a 
very fine sieve, and mixed up, as it is Avanted for use, with Flan- 
ders glue, isinglass, or some other material of the same kind. This 
solution is colored with the tint required for the scagliola, but 
when a marble of various colors is to be imitated, the several 
colored compositions required by the artist must be placed in 
separate vessels, and they are then mingled together in nearly the 
same manner that the painter mixes his color on the pallet. 
Having the wall or column prepared with rough plaster, it is 
covered with the composition, and the colors intended to imitate 
tlie marble, of whatever kind it may be, are applied when the 
floating is going on. 

It now only remains to polish the work, which, as soon as the 
composition is hard enough, is done b}^ rubbing it Avith pumice- 
stone, the work being kept wet with water applied by a sponge. 
It is then polished with Tripoli and charcoal, with a piece of tine 
linen, and finisiied with a i:)iece of felt, dipped in a mixture of oil 
and Tripoli, and afterwards with pure oil. 



304 Plastering. 



Co7npositio7i. 

This is frequently used, instead of plaster of Paris, for the orna- 
mental parts of buildings, as it is more durable, and becomes in 
time as hard as stone itself It is of great use in the execution of 
the decorative parts of architecture, and also in the finishings of 
picture frames, being a cheaper method than carving, by nearly 
eighty per cent. 

It is made as follows: Two pounds of the be*', whitening, one 
pound of glue, and half a pound of linseed oil are heated together, 
the composition being continually stirred until the different sub- 
stances are thoroughly incorporated. Let the compound cool, and 
then lay it on a stone covered with powdered whitening, and heat 
it well until it becomes of a tough and firm consistence. It may 
then be put by for use, covered with wet cloths to keep it fresh. 
When wanted for use it must be cut into pieces, adapted to the 
size of the mould, into which it is forced by a screw press. The 
ornament, or cornice, is fixed to the frame or wall with glue, or 
with white lead. 

To make Glass Paper. 

Take any quantity of broken glass (that with a greenish hue is 
the best), and pound it in an iron mortar. Then take several sheets 
of paper, and cover them evenly with a thin coat of glue, and, 
holding them to the fire, or placing them upon a hot piece of wood 
or plate of iron, sift the pounded glass over them. Let the several 
sheets remain till the glue is set, and shake off the superfluous 
powder, which will do again. Then hang up the papers to dry 
and harden. Paper made in this manner is much superior to that 
generally purchased at the shops, which chiefly consists of fine 
sand. To obtain different degrees of fineness, sieves of different 
degrees of fineness must be used. 

To make Stone Paper. 

As, in cleaning wood-work, particularly deal and other soft 
woods, one process is sometimes found to answer better than 
another, we may describe the manner of manufacturing a stone 
paper, which, in some cases, will be preferred to sand paper, as it 
produces a good face, and is less liable to scratch the work. Having 
prepared the paper as already described, take any quantity of 
powdered pumice-stone, and sift it over the paper through a sieve 
of moderate fineness. AVhen the surface has hardened, repeat the 
process till a tolerably thick coat has been formed upon the paper, 
which, when dry, will be fit for use. 



Woodwork, CarpentPwY, etc. SOS 



l_ 



WOODWOEK, CAEPENTRY, &c. 

Decay of Wood. 

Some woods decay much more rapidly than others; but they 
will all, in some situations, lose their fibrous texture, and with it 
their properties. To ascertain the causes which act upon woods, 
and effect their destruction, is an important object both to the 
builder and to the public. 

Cause of the Decay of Thnher, 

All vegetable as well as animal substances, when deprived of 
life, are subject to decay. 

If the trunk or branch of a tree be cut horizontally it will be 
seen that it consists of a series of concentric layers, differing from 
each other in color and tenacity. In distinct species of trees these 
layers present very different appearances, but in all cases the outer 
rings are more porous and softer than the interior. Wood is essen- 
tially made up of vessels and cells, and the only solid parts are 
tliose coats which form them. These vessels carry the sap which 
circulates through the tree, gives life and energy to its existence, 
and is the cause of tlie formation of leaves, flowers, and fruit. But 
when the tree is dead, and the sap is still in the wood, it hecomea 
the cause of vegetable decomposition by the process of fermentation. 
There are five distinct species of vegetable fermentation — the 
saccharine, the coloring, the vinous, the acetous, and the putrefac- 
tive. We are indebted to Mr. Kyan for the discovery that albumen 
is the cause of putrefactive fermentation, and the subsequent de- 
composition of vegetable matter. 

Circumstances favorable to Vegetable Decomposition, 

Wood is not equally liable to decay under all circumstances. 
When thoroughly dried it is not so quickly decomposed as when in 
its green state, for in the latter condition it has in itself all the 
elements of destruction, and it is scarcely possible to prevent the 
effect if it be then used in building. But supposing the timber to 
be perfectly seasoned, it is more liable to decay under some circum- 
stances than in others. Timber is most durable when used in very 
dry places. 

When timber is constantly exposed to the action of water, tlie 
decomposition effected will depend upon the nature and chemical 
composition of the substance. A portion of wood ma}' be soluble 
in water, but other parts are not ; so that after a definite period, 
the continued action of water upon a piece of timber ceases, and if 
it can sustain the influence of this cause until that period tliere is 
no termination to its endurance, except from those casualties which 
it might have been able to bear in its original state, but cannot 
after the removal of tliat portion of its substance soluble in water. 

_ 



306 Woodwork, Carpentry, etc. 



I. 



Should a piece of timher thai has been for a long tirne exposed to water 
he brought into the air and dried, it will become brittle and useless. 

When wood is alternately exposed to the influence of dryness 
and moisture it decays rapidly. It appears, from experiments, that 
after all the matter usually soluble in water has been removed, a 
fresh maceration and contact of the air produces a state of matter 
in that which is left which renders it capable of solution. A piece 
of timber may then in this manner be more and more decomposed, 
until at last the whole mass is destroyed. The builder is sometimes 
compelled to use wood in places where it will be exposed to alter- 
nate dryness and moisture ; fencing, weather boarding, and other 
works, are thus exposed. In all these cases he may anticipate the 
destructive process, and provide against it. The wood used in such 
situations should be thoroughly seasoned, and then painted or 
tarred; but, if it be painted ichen iiot thoroughly seasoned, the 
DESTRUCTION WILL BE HASTENED, for the evaporation of the contained 
vegetable juices is prevented. 

There is one other circumstance to be considered — the influence 
of moisture associated with heat Within certain limits the ' 
decomposition resulting from moisture increases with the tempe- 
rature. The access of the air is not absolutely necessary to the 
carrying on of this process, but water is ; and as it goes on, car- , 
bonic acid gas and hydrogen gas are given off. The woody fibre ; 
itself is not free fi'om this decomposition, for, as the carbonaceous \ 
matter is abstracted by fermentation, it becomes more susce]3tible I 
of this change. This statement is proved by the circumstance, that [ 
when quicklime is added to the moisture the decomposition isj 
accelerated, for it abstracts carbon ; but the carbonate of lime pro- j 
duces no such effect: a practical lesson may be learnt from this 
fact ; if timbers be bedded in mortar, decomposition must follow, 
for it is a long time before it can absorb sutficient carbonic acid to 
neutralize the effect, and the dampness which is collected by con- 
tact with 4:he wet mortar increases the effect. When the wood and 
the lime are both in a dry state no injury results, and it is well 
known that lime protects wood from worms. 

When the destructive process first becomes visible it is by the 
swelling of the timber, and the formation of a mould or fungus 
upon its surface. This fungus or cryptogamic plant rapidly in- 
creases, and soon covers over the whole surface of a piece of timber, 
having a white, grapsh-white, or brownish hue. When the seeds 
of destruction are thus once sown they cannot be readily eradicated. 
Heat and moisture mav be considered the prominent causes of the 
rapid decomposition of vegetable substances. When wood is com- 
pletely and constantly covered with water this effect is not pro- 
duced ; and we have an example in the fact, that, although those 
parts of a vessel which are subject to an occasional moisture are 
liable to dry rot, yet those parts which are constantly beneath the | 
water are not ever thus affected ; and although the head Qf a pile, \ 
which may be now and then wetted by the casual rise of the tide, i 



4 



"Woodwork, Carpentry, etc. 307 



and is then dried again by the sun, may be decomposed, yet tJioi^e i 
parts which are always covered with water have been found in a solid \ 
slate after centuries of immersion. | 

Means of Preventing Decay. 

Something may be done towards the prevention of decay by 
felling the timber at a proper season. A tree may be felled too 
soon or too late, in relation lo its age and to the period of the year. 
A tree may be so yonng that no part of it shall have the proper 
degree of hardness, and even its heart-wood may be no better than 
sap-wood; or a tree may be felled when it is so old that the wood, 
if not decayed, may have become brittle, losing all the elasticity of 
maturity. The time required to bring the several kinds of trees to 
maturity varies according to the nature of the tree and the situa- 
tion in which it may be growing. Authors differ a century as to 
the age at which oak should be felled, some say one hundred, and 
others two hundred years ; it must, then, be regulated according 
to circumstances. 

But it is also necessary that the timber trees should be felled at 
a proper season of the year ; that is to say, when tlieii* vessels are 
least loaded with those juices which are ready for the production 
of sap-wood and foliage. TJie timber of a tree felled i7i sprioig or iit 
autumn woidd be especially liable to decay ; for it would contain the 
element of decomposition. Midsummer and midAvinter are the 
proper times for cutting, as the vegetative powers are then ex- 
pended. 

There are gome trees, the bark of which is valuable, as well as 
the timber ; and as the best time for felling is not the best for 
stripping the bark, it is customary to perform these labors at 
different periods. The oak-bark, for instance, is generally taken off 
in early spring, and the timber is felled as soon as the foliage is 
DEAD ; and this method is found to be highly advantageous to the dnra- 
bility of the timber. The sap-wood is hardened, and all the avail- 
able vegetable juices are expended in the production of foliage. 
Could this plan be adopted with other trees, it would be desirable; 
but the barks are not sufficiently valuable to pay the expense of 
stripping. 

Seasoning Timber. 

Supposing all these precautions to be taken in felling timber, it 
is still necessary to season it ; that is, to adopt some means by 
which it may be dried, so as to throw off all the juices which are 
still associated with the fibres of the wood. As soon as the timber 
is felled, it should be removed to some dry place ; and, being piled 
in such a manner as to admit a circulation of air, remain in log for 
some time, as it has a tendency to prevent wai'ping. The next 
process is to cut the timber into scantlings, and to place these 
upright in some dry situation, Avhere there is a good current of 
air, avoiding the direct rays of the sun. The more graduall}^ the 



308 "Woodwork, Carpentry, etc. 



process of seasoning is carried on, the better will be the wood for 
all the purposes of building. Mr. Tredgold says, " It is well known 
to chemists, that slow drying will render many bodies less easy to 
dissolve ; while rapid drying, on the contrary, renders the same 
bodies more soluble. Besides, all wood, in dr^'ing, loses a portion 
of its carbon, and the more in proportion as the temperature is 
liigher. There is in wood that has been properly seasoned a 
toughness and elasticity which is not to be found in rapidly dried 
wood. This is an evident proof that firm cohesion does not take 
place when the moisture is dissipated in a high heat. Also, sea- 
soning by heat alone, produces a hard crust on the surface, which 
will scarcely permit the moisture to evaporate from the internal 
part, and is very injurious to the wood. 

"For the general purposes of carpentry, timber should not he used 
in less than two years after it is felled ; and this is the least time 
that ought to be allowed for seasoning. For joiners' work it 
requires four years, unless other methods be used; but, for car- 
pentry, natural seasoning should have the preference, unless the 
pressure of the air be removed." 

Many artificial methods of seasoning timber have been proposed ; 
and a brief notice of some of those which have been found most 
useful will be required. 

Seasoning hy a Vacuum, 
All the vegetable and animal juices are kept in their particular 
vessels by the pressure of the atmosphere : remove that pressure, 
and the animal fluids could no longer be retained by the veins and 
arteries ; and the vegetable fluids would exude and appear on the 
surface of the plant. Place a small piece of wood beneath the 
receiver of an air-pump, and exhaust the air, and in a short time 
the wood will be covered with drops of the liquid which can no 
longer be retained, as the atmospheric pressure is removed. Mr. 
Langton thought that this might be applied to the extraction of 
those vegetable juices in timber, known to be the cause of its 
decay. An arrangement was therefore adopted, by which large 
masses of timber might be inclosed in a vessel having such ma- 
chinery as would be necessary to exhaust the air, heat being at 
the same time employed so as to vaporize the exuded juices. The 
vapor is conveyed away by pipes surrounded by cold water, and is 
condensed into liquid having a sweet taste. This process is deserv- 
ing of more attention than has hitherto been given to it. 

Water Seasoning. 

It has been stated, by various writers, that wood immersed in 
water for about a fortnight, and then dried, is better suited for all 
the purposes of the joiner. There can be no doubt that immersion 
in water tends to neutralize the eff'ect of the saccharine matter, by 
dilution or an almost absolute removal. This process has also the 
effect of rendering the wood less liable to crack and warp ; but, if 



Woodwork, Carpentry, etc. 300 



we judge by Duhamel's experiments, it injures tlie strength of tlie 
material, and should not, therefore, be adopted in any instance 
where the timber is to be employed by the carpenter. Evelyn 
recommends boards that are to be used for flooring to be seasoned 
in this way : " Lay your boards," he says, " a fortnight in water, 
(if running the better, as at a mill-pond head;) and then setting 
them upright in the sun and wind, so as it may pass freely through 
them, turn them daily ; and thus treated, even newly-sawn boards 
will floor far better than those of a many years' dry seasoning, as 
they call it." Timber intended for ship building may be immersed 
in sea water ; but that which is to be used for //oii.se.s ought to be 
placed in fresh water ; for if timber, or any other building material, 
be impregnated with salt, it will ever be wet, for salt attracts 
moisture so readily that it may be used approximately as a hygro- 
meter. Plaster or mortar made with salt water will ahvays sic^at 
with a moist atmosphere ; and timber intended for the house car- 
penter, if impregnated with salt, w^ill always be damp, or covered 
with a crystallized efflorescence. Much injury, however, is some- 
times done by not thoroiigldy immersing the timber; the carpenter 
should therefore be careful when he employs this method of season- 
ing, that the timber is entirdy covered with water, and that it be 
not exposed to its action for too long a time. 

Seasoning by Smoking a?id Charring. 

Authors who have written upon the seasoning of timber have 
spoken of the efl'ects of smoke, and the carbonization of the surface. 
We have adopted the same arrangement, but it will be necessary 
to caution the reader against a misconception of a very inaccurate 
expression. Timber cannot be seasoned by either smoking or 
charring, but seasoned timbers may be made more capable of 
resisting the effects of certain situations by these processes. Should 
a piece of timber, containing the vegetable juices, be smoked or 
charred, it would be a means of accelerating decomposition ; for 
preventing all means of evaporation, the common sources of protec- 
tion would become sources of destruction. But when timber is to 
be used in situations where it is liable to be attacked by worms, or 
to produce fungi, it may be desirable to smoke or to char it. 

Seasoning by Boiling or Steaining, 

Timber is sometimes seasoned by steaming or boilino-, both of 
which means are frequently adopted b}^ ship-builders. Tlie' strength 
of timber appears to be somewhat impaired by these processes, but 
it is generally less liable to shrink or crack. Duhamel states that 
he boiled a piece of wood, and then dried it upon a stove, but in 
drying it, itlost part of its substance, as well as the water contained, 
and, upon a repetition, he found that it had lost still more of its 
weight. Four hours' exposure to steam or boiling water is sufTicient 
for timbers of ordinary dimensions, and the drying afterwards goes 



310 Woodwork, Carpentry, etc. 

on very rapidly, but it should be done as gradually as possible. 
The joiner frequently finds it necessary to steam or boil wood, to 
bend it into a particular curve, and also the ship-builder. It has 
been stated by writers on ship-building, that boiling increases the 
durability of timber; and, in proof of this, they inform us that the 
planks in the boAV of a ship, which are bent in this way, are never 
affected by the dry rot. 

It may uow be inquired whether, after the most perfect season- 
ing, timber is secured against tbe process of decay ? To this ques- 
tion a negative answer must be given. However well the timber 
ma}' be seasoned, it will certainly rot if placed in a damp situation, 
the rapidity of the decomposition depending upon the nature and 
state of the wood, and the activity of tlie destroying agent. As the 
builder seldom attempts any other seasoning than that which 
depends upon dr\ing his timbers, it is absolutely necessary that lie 
should carefully avoid the rise of damp, and adopt every means in 
liis power to prevent this evil. Timbers are usually placed in con- 
tact with walls, but it must not be supposed that this is sufficient 
to keep them from the access of damp, for they are frequently the 
conducting media. Brickwork very readily absorbs moisture, and 
also throws it upwards, so that the ends of timbers are in contact 
with the very source of mischief. To prevent the rise of damp 
upwards, it is common to use, for a few feet above the foundations, 
cement, a substance impei'vious to water, instead of mortar, or to 
place between the courses zinc or slate. But that these plans may 
be effective, the basement walls should be surrounded with an open 
area, for, if in contact with the earth on their sides, they can be of 
no value. To prevent dampness from entering in front, the brick- 
work should be covered with compo, or some substance imper- 
meable to water. 

Another thing to be considered, for the security of timbers, is to 
arrange, in every plan of a building, for a perfect circulation of air. 
Ventilation is a most important requisite in the construction of a 
building, although it is generally a matter of very little impoi-fance 
in the consideration of those who have to plan or construct build- 
ings. The ventilation of roofs is by no means difficult, but there 
are often so many obstacles to the ventilation of flooring that the 
designer will not give sufficient attention to his subject to provide 
against them. These things, however, are not matters of specu- 
lation, to be attended to by those who have no higher employment, 
but are absolutely necessary for the construction of a work that is 
intended to survive the builder. 

The attention of scientific men has been recently directed to the 
experiments made by Mr. Kyan. Having made a great number of 
experiments with a view to ascertain the primary cause of vegetable 
decomposition, he was at last convinced that albumen was that 
cause, and that to neutralize its effects would be to prevent decom- 
position. Some plan was required similar to that adopted in 
tanning. The gelatin in animal bodies is quite as liable to decom- 



Woodwork, Carpentry, etc. 311 



position as the albumen of vegetables; but when tannin, the infu- 
sion of oak bark, is combined with it, the destructive properties are 
lost, and the animal matter becomes durable, and almost incanable 
of decay. Reasoning upon this effect, Mr. Kyan imagined that it 
might be possible to prevent vegetable decomposition by causing 
the albumen to form a combination with some other substance; 
and, knowing the affinity of corrosive sublimate for the albumen, 
he entered upon a series of experiments, which led him to propose 
the use of that substance as a protection for timber. 

Mr. Kyan inferred that, as wood consists of various successive 
layers, in which the albumen, or juices containing albumen, circu- 
lated freely, it is quite certain, as these juices withui the wood, 
with the watery parts, fly off by the leaves, that the albumen 
remains behind, and it is probable that this albumen, which from 
its nature is peculiarly prone to enter into new combinations, is the 
thing in wood which begins the tendency to decomposition, and 
produces ultimate decay, whether that decomposition is attended 
with the formation of cryptogamic substances, or whether in the 
less organized form, the change occurs with the simple production 
ot what has been called the dry rot. Mr. K. conceived, therefore, 
if albumen made a part of wood, the latter would be protected by 
converting that albumen into a compound of protochloride of 
mercury and albumen; and he proceeded to immerse pieces of 
wood in this solution, and obtained the same result as that which 
he had ascertained with regard to the vegetable decoctions. 
Having done so, it became necessary to employ various modes of 
experiment, as well as comparative experiments. IS'ow it is not 
clear in what part of the wood the vegetable albumen may be 
found, though it exists more especially in that part of the tree 
which is denominated the alburnum or sap, and is found between 
the heart-wood and the innermost layer of bark. The experience 
of all practical men has confirmed the opinion, that this portion of 
wood is the first to decay. 

It is probable that, as the alburnum becomes successive layers of 
wood, it loses a quantity of albumen; or that, in consequence of 
the pressure which takes place by the addition of each successive 
layer, it becomes so situated as to lose a part of its exposure to the 
vessels where a change may occur, and therefore becomes, in some 
measure, protected; for that which is one year alburnum or sap, 
may be, and indeed generally is, proper w^ood the next. 

The mode in which the application of the solution takes place is 
in tanks, wliich may be constructed of different dimensions, from 
twenty to eighty feet in length, six to ten in breadth, and three to 
eight in depth. The timber to be prepared is placed in the tank, 
and secured by a cross-beam to prevent its rising to tlie surface. 
The wood being thus secured, the solution is then admitted from 
the cistern above, and for a time all remains perfectly still. In the 
course often or twelve hours, the water is thrown into great agita- 
tion by the effervescence occasioned by the expulsion of the air 



J 12 "Woodwork, Carpentry, etc. 



fixed in the wood, by tlie force witli whicli the fluid is drawn in bj 
cliemical affinity, and by the escape of that portion of the chlorine, 
or muriatic acid gas, which is disengaged dui-ing the process. In 
the course of twel^'e hours this commotion ceases, and in the space 
of seven to fourteen days, var3-ing according to the diameter of the 
wood, the change is complete, so that as the corrosive sublimate is 
not an expensive article, the albumen may be converted into an 
indecomposable substance at a very moderate rate, and the season- 
ing will take place in the course of two or three weeks." 

Mr. Kyan's method of seasoning has been already tested, under 
circumstances so severe, that they may be said to have proved its 
cfficienc}^ A piece of oak was five years in the fungus pit in Wool- 
wich yard, London, a place notorious for the rapid and almost 
instantaneous destruction of vegetable matter, and it was as sound 
when taken out as when put in. This was the most severe test to 
which the method could be subjected, and its having sustained the 
trial is a proof of the value of the discovery. It has, however, been 
objected to the process, that the impregnation of timber with corro- 
sive sublimate must unfit it for use in ship-building ; but Mr. Kyan 
has furnished evidence to the contrary, and proves that salubrity is 
one advanta2^e. We strongly recommend the builder to make 
experiments himself upon wood prepared by Mr. Kyan, by using it 
in places where decay is rapid. 

Framing of Timbers. 

"When timbers are framed together, it is with the intention of 
supporting some weight, or resisting the strains to which the mate- 
rials may be exposed in the situations where they are to be placed. 
Horizontal or vertical timbers are not alwa3's of themselves suffi- 
ciently strong to sustain the pressure to which they may be sub- 
ject, but they need assistance, and it then becomes a question, how 
can the materials intended to assist be best applied, and what are 
the smallest scantlings that can be adapted ? Two things must be 
studied — stability and economy. It has been often stated that these 
two results cannot be accomplished by the same arrangement, but 
as the forces which are to be opposed have usually a direct appli- 
cation, so the system by whicli they are to be resisted may, 
usually, be of a simple construction. 

Composition and Resolution of Forces. 

Two great mechanical principles lie at the base of all proper 
attempts to estimate the nature of the forces which may be exerted 
upon substances in particular situations ; these principles are called 
tlie composition and the resolution of forces. 

The resolution of forces is the means of finding any two or more 
forces which may resist or control the pressure of any one force. 
The composition of forces consists in finding the direction and 
amount of one force that is capable of producing the same effect as 



Woodwork, Carpentry, etc. 



313 



two or more forces acting in different directions. This is, in fact, j 
only the reverse of the resolution of forces, and the two are, 
strictly speaking, but one principle; and if the one process be| 
understood, the other must be almost so of necessity. Nor may the! 
student pass over this part of the work, under a fear that it is tooi 
mnthematical for liim to understand,. for he can never be certain | 
that the roofs or other framing which lie may design will support; 
the Aveights they are intended to carry, if he does not know how to! 
calculate the action of the weights or forces by which they may be 
pressed. 

Let B D, fig. 1, be the king-post of a roof, and let BA, B C, be 




Fig. 1. 



I 



the rafters: they are framed together for the purpose of carrying 
some weight; and the (piestion is this — are they sufficiently strong! 
to carry the weight which is to be placed upon them ? To deter- 1 
mine this we must refer to the resolution of forces. Let us sup- 1 
pose some determined weight to rest upon the point B. Then, by | 
some scale of equal parts, draw a line B d, equal to the number of j 
pounds, hundred weights, or tons, resting upon the point B, and I 
draw da parallel to B C, and dc parallel to B A. Now measure the i 
line a B by the same scale, and it will give the number of pounds, 
hundred weights, or tons, by Avhich A B is strained, and c B will 
give the strain upon B C. But, in the drawing affixed, the rafter 
B C is longer than the rafter B A ; but this does not at all affect 
the weight, for it remains the same, whatever may be the length of 
the beam which carries it ; but it is necessary to remember that, 
by increasing the length of the beam, it is rendered less capable of 
supporting the weight, and a proportionate increase of dimensions 
must be allowed. But should the direction of the beam be 
changed, a very different result will be obtained, for in every case 
tlie pressure will be increased or decreased. The strain upon the 
beam B A, fig. 2, Avill now be measured by the line a 6, and that 
upon B c by 6 c. In fact, a very slight alteration of position may, 
under certain circumstances, enormously increase or decrease a 
strain. It will be scarcely necessary to explain how two or more 



27 



814 



Woodwork, Carpentry, etc. 




forces may be com- 
posed, and the sin- 
gle force, acting in 
a certain direction, 
be calculated. 

Leaving the sub- 
ject of the compo- 
sition and resolu- 
tion of forces, after 
a statement of the 
principle, we may 
proceed to explain 
the construction 
and arrangement 
of those parts of a 
building which be- 
long to the carpenter. And, first of all, we may speak of roofs. 

The Construction of Roofs, 

The simplest method of constructing a roof is to place horizontal 
timbers from wall to wall, but this method is only suited to very 
short bearings, and does not readily throw off the water which 
may fall upon its covering. The Egyptians constructed flat roofs. 
To prevent this inconvenience, a roof ma}' be made as an inclined 
plane ; and such a construction has advantages, though its want 
of uniformity and beauty, and also its want of strength, propor- 
tioned to the amount of timber employed, are objections to its use; 
but still it is stronger than the flat roof, and readih' carries off the 
water that may fall upon it. Tlie best form for a roof is that in 
which there are two sides, equally inclined to the horizon, and 
resting in a line called the ridge of the roof The angle which the 
inclined side forms with the horizon is called the pitch. In coun- 
tries where there is a cold climate, and snow is apt to fall in large 
quantities, the roof is high ; in warm countries the roof is low. In 
Gothic architecture the roof is generally high pitched, and it is so 
consonant with the style that it often forms a prominent feature in 
these buildings. There are not so many advantages in high pitched 
roofs as most persons suppose, and there are many disadvantages. 
The additional force of the wind upon a high roof is a serious 
objection, and when parapets are emploved it is so far from pre- 
venting the eftects of a heavy fall of rain or snow that the gutters 
are so filled that the pipes cannot carry off the water fast enough, 
or, being stopped by the dirt carried d"own by the velocity of the \ 
water, an overflow is occasioned. The height of roofs is now gene- 1 
rally between one third and one sixth of the span. i 

It is the carpenter's business to frame the timbers of roofs, and i 
sometimes he is required to design them,, and he should therefore j 
know how to obtain the strength and other qualities required, with j 
the smallest possible amount of timber. \ 



Woodwork, Carpentry, etc. 



315 



A piece of timber, in whatever way it may be placed, except 
when vertical, w^ill bend or sag, that is to say, its upper side will 
form itself into a concave surface. The more horizontal the timber 
is placed the more it will always sag, and as the distance between 
the points on which it rests is increased, so it has greater liabilities 
of bending. To prevent this effect as much as possible, arrange- 
ments must be made for the support of the beam in some inter- 
mediate points. IS'ow, it may be supported from either above or 
below. If there should be any walls between those on which the 
ends of the timber rest, these will be sufficient for all the purposes 
required ; if not, the same result must be produced by a system of 
framing. 

The timbers which compose a roof are knoAvn by different names, 
according to the uses for which the}' are emplo3'ed, and the situa- 
tions in Avhich they are placed. The principal timbers of a roof are 
the following, but they are not all used in every roof: the tie- 
beams, wall-plates, collar-beams, king-posts, queen-posts, struts, 
principal rafters, common rafters, ridge-piece, collar-beams, purlins, 
and pole-plates. 

The Tie-beam (a), fig. 3, is a horizontal piece of timber, which 




Fig. 3. 

extends from wall to wall, and rests upon the Wall-plates (b) at 
each end. It is employed for the purpose of connecting tlio feet of 
the principal rafters (C), which would otherwise have a tendency to 
push out the walls by their own weight, and the weight of the 
materials placed upon them. In roofs of large span, it is necessary 
that tlie tie-beam should be well supported in some point or 
points, between the ends on wliich it is supported, for if this be not 
done it will sag and draw either one or both of the principal raf- 
ters towards its centre, and thus destroy the stability of the 
framing. The King-post (D) is sometimes used for this purpose. It 



316 



Woodwork, Carpentry, etc. 



is a piece of timber placed in a vertical position, connecting the 
point where the two principal rafters meet, and the centre of the 
tie-beam. 

When the king-post is not thought to be sufficient to support the 
pressure which may be on the framing. Queen-posts (B), fig. 4, may 
be used, which are pieces of timber placed in an upright position, 




Pig. 4. 

supporting severally the two rafters, and equidistant from the 
centre of the truss. The horizontal piece of timber (C) which con- 
nects the heads of the queen-posts, is called a straining-beam; and 
that which connects their base, so as to prevent the struts from 
pushing them nearer to each other, is called a straining cill. Those 
pieces which are placed in j^airs, to assist in supporting the prin- 
cipal rafters, are called struts ; they are frequently used to unite 
the rafters and the base of the king-post. Any horizontal timber 
above the tie-beam is called a collar-beam. The ridge-piece (H) 
is that piece of timber which forms the apex of the roof, and is 
supported by the heads of the principal rafters or the king-posts, 
and in its turn supports one end of the common rafters. A pole- 
plate is a beam over the walls, supported by the principal rafters 
or the tie-beam, and is intended to carry the lower ends of the 
common rafters. Purlins (E) are horizontal timbers, between the 
pole-plates and ridge-piece. The small spars (c c), which are 
parallel to the principal rafters, and are supported by the ridge- 
plate, purlins, and pole-plates, are called common rafters. 

The Dimensions of Timbers used in a Roof. 

However accurately a roof may be designed, it is unfit for its 
purpose if the dimensions of the parts be not accurately propor- 
tioned. To accomplish this, some experience is required, and a 



Woodwork, Carpentry, etc. 



317 



knowledge of the strength of timbers, under particular circum- 
stances. 

There are two things to be secured— a sufficient strength to sup- 
port the weights to be carried without sagging, and to do that 
witliout burdening the walls or other parts of the building over 
wliich the roof is thrown. This is not always an easy task, for 
roofs are sometimes to be made in such forms as prevent the adop- 
tion of those means which would otherwise immediately accomplish 
the object. Sometimes a very large roof must be made flat, at other 
times a lantern-light must "be provided in its centre ; and, in a 
third case, it may be necessary to erect a dome. In designing for 
these and other roofs, attention should be paid to the character and 
success of similar works already executed, and the artist should 
study the points of similarity and difference between these and his 
own* work, so as to provide against dangers, which may peculiarly 
aftect his building. 



Examples of Roofs. 

Fig. 5 is a roof, the raf- 
ters of which are only sup- 
ported by a collar-beam (C), 
which acts in part as a tie ; 
but this arrangement is so 
feeble, that it should never 
be used over a space where 
the span is more than fifteen 
feet. 



In fig. 6 there is the addition of a tie-beam (A), and the strain is 
here thrown from the collar to the tie-beam; the former being 
compressed, the latter in a state of tension. As there is no arrange- 





ment in this truss to support the tie-beam, and to prevent it from 
sagging, it is unfit for a span of more than twenty-five feet. 



318 



Woodwork, Carpentry, etc. 



To prevent the inconveniences resulting from the sagging of the 
tie-beam, a king-post (P) and struts (SS) may be introduced, as 




shown in fig. 1. This form of roof is very well adapted for a span 
of twenty-five feet. 
For a span of thirty to five-and-forty feet, the truss represented 



^1 1^ 





Fig. 8. 

in fig. 8 is very well suited, and is now very commonly adopted by 
architects and builders. 

Floors, 

The timbers which support the flooring boards, and the ceiling 
of a room beneath, are called, in carpentry, the naked flooring. 

There are three kinds of naked flooring — single, double, and 
framed. 

Single flooring is that in which there is but one series of joists, as 
shown in fig. 9, where AAA are joists, and B the flooring-boards. 
To make a single floor as strong as possible, the joists should be 
thin but deep, sufl[icient thickness being always allowed for the 
nailing of the flooring boards. Two inches by six is the smallest 



Woodwork, Carpentry, etc. 



319 




Fig. 9. 
dimension for joists ; for a length of twenty feet they should be 
about three inches thick, and twelve inches deep. 

Sometimes the joists cannot have in a particular place a bearing 
upon the walls, and then a piece of timber is framed between the 
nearest joists. This is done where flues, fire-places, and stairs inter- 
fere. The timber thus used is called a trimmer, and the two joists 
on which it is supported are called trimming-joists, and should be 
made a little stronger than the common joists. Thus, in fig. 10, 




Fi?. 10. 

A A are common joists, BB trimming joists, and C a trimmer. 
When the bearing is more than seven or eight feet, the joists 
should be strutted ; that is to say, short pieces of board should be 
fitted between the joists, so as to form a continued line from w\all 
to wall. These struts greatly strengthen the floor, and prevent the 
joists from sinking; but it is not desirable to mortice them into the 
joists, as that process has the efl'ect of weakening the joists them- 
selves. 




Fig. 11. 



320 



"Woodwork, Carpentry, etc. 



Double flooring is that in which there are two tiers of joists, the 
binding joists, as A A, in fig. 11, which in fact support the floor, and 
the bridging joists B B. In this kind of flooring, the binders extend 
from wall to wall, and the bridging joists are notched down upon 
them. Beneath the binders we have a third tier of timbers (D), 
which are pulley-morticed into the binders, and are called ceiling 
joists. 

When the binding joists are framed into a large piece of timber, 
called a girder, the floor is said to be a double framed floor. Thus 
in fig. 12 A is the girder, B a binding joist, C a bridging joist, D D 




Fig 12. 

ceiling joists, and E flooring boards. This kind of floor is deci- 
dedly the best when it is necessary to provide for a good and even 
ceiling, for although single floors may be made very strong for a 
great bearing, yet the ceilings are always liable to crack. 

It is not easy to obtain timber for girders of much more than 
twenty feet scantling, and they are therefore trussed. Trusses are 
used in both floors and roofs, but we have not thought it desirable 
to interrupt the course of explanation we have given, by a refer- 
ence to any particulars concerning this branch of carpenter's 
work; yet it is necessary that we should now make a few remarks 
upon it. 

Trusses. 

When timbers are so framed together as to support weights, 
they are called trusses. It frequently happens that a piece of 
timber, in itself incapable of supporting a weight, may, when cut 
into scantlings of difl"erent dimensions, and framed together, not 
only carry that weight, but also support a much greater. The bow 
and string roof, invented by Mr. Smart, is an example in point. 



c 


o 


5 


\l 


i 


\y 


a." 


s 


€ 



Fig. 13. 



"Woodwork, Carpentry, etc. 



321 




Fiff. 14. 



Let A A, in fig. 13, be a piece of timber, wliich we will suppose to 
be insufficient of itself to carry a particular weight ; from this cut 
the pieces o, s, e, 6, and o, s, d^ c. Then let these pieces be raised 
as in fig. 14, and a key be placed between them at the apex ; and 
it will form a very strong 
truss, which may be made 
still more capable of resisting 
a strain, by the application of 
struts.. 

The principal rafters of a 
roof are so called because they are trussed. It is not necessary to 
truss all the rafters in a roof, and it would be very expensive to do 
so ; and therefore trusses are placed at particular distances from 
each other, according to the weight to be carried ; and they are 
formed in different ways, according to the span over which they 
are thrown. 

It has been already stated, that girders are sometimes trussed, 
and should always be when their bearing is much more than 
twenty feet. We have often seen trusses which, so far from 
strengthening the girders, have decidedly weakened them. Large 
girders are sometimes sawn down the middle, and when reversed, 
are bolted together with slips of wood between them. It has 
been supposed that this strengthens, and is adopted for this par- 
pose ; but the supposition is erroneous, though the plan is certainly 
a good one, for it allows a free circulation of air between the 
pieces, and facilitates the emission of any dampness that may be in 
the timber. 

A strong girder may be made as strong, in fact, as any truss of 
the same deptli, by bolting two pieces of timber together, or by 
confining them wi'th iron hoops, the ends of the girder being 
smaller than the centre, so as to allow the hoops to be driven 
tighter, and confine the beams. 

In fig. 15 we have given a representation of a strong truss 



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Fig. 15. 

girder, the truss post and the abutment pieces being made of 
wrought iron. 

Of Connecting Timbers. 

It is sometimes impossible to obtain timbers of the length 
required for the several parts of a building, and it is then necessary 
to join two or more pieces together, so as to form them into one 
piece, and to injure the stability as little as possible. This process 



Woodwork, Carpentry, etc. 



is called scarfing, and the parts of the joints 'which conao in contact! 
are called scarfs, and are usually connected by iron bolts. J 

There are many ways of soarliug, every builder adopting that' 
one which appeai-s to him the best under the circumstances in> 
which the timber is to be employed. Two or three different! 
methods may be mentioned, leaving the workman to examine those! 
which he may happen to meet with in practice, and the various! 
designs which have been given by writers on the art of building. [ 

Fig. 16 shows the means of searling without diminishing the| 



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Fig. IS. 



length of the pieces. Tliis is done by the introduction of a third i 
piece, havinsr the form of steps, and all the pieces being united; 
together by bolts and plates. ! 

Fig. ] 7 is a representation of a scarfing, which is very simple, ' 
and frequently used, though there is a considei^ble loss of timber. ' 



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' Tlie pieces to be united are connected by iron bolts, an iron plate 
being placed on both sides. 

Fig. IS represents a form of scarfing, adapted to a beam, which; 




Fig. 13. 



has to support a cross strain. In many arrangements, the whole 
strain is supported by the straps and boks, but in this they do not, 
in consequence of the indentation. ' j 

Timber Partition&. 

liooms and passages are often separated by timber partitions^ | 

which are so formed as to be covered with lath and plaster. In^ 

fig, 19 we have given a design for the framing of a partition, with- 

a door through it: A A are^the door-posts, B the head, C the sill,' 

I DD are braces which support the quartering, and are assisted by| 



Woodwork, Cakpentry, ktc. 



323 





c 



Fig. 19. 

the struts, EE. It will be quite evident from a glance at the draw- 
ing, that the door-posts help to sustain the braces and struts; while 
they in return prevent the fall of the door-posts. Braces may be 
introduced in various ways, but strength is the object for which 
they ought to be introduced, a circumstance which is very fre- 
quently entirely forgotten by carpenters. In some instances, it 
may be found desirable to introduce a simple truss into a design 
for partitions. 

The carpenter usually connects his timbers either by notching, or 
by mortice and tenon. Dovetail joints are sometimes used in car- 
pentry, but they ought never to be adopted, for they will always 
draw when the timber shrinks, and the oblique surface of the 
dovetail tends to force the timbers apart, acting as though it were 
a wedge. 

Gluing Joints. 
In general, nothing more is necessary to glue a joint, after the 
joint is made perfectly straight, or, in technical terms, out of 
winding, than to glue both edges Avhile the glue is quite hot, and 
rub them lengthwise until it has nearly set. When the wood is 
spongy, or sucks up the glue, another method must be adopted, one 
which strengthens the joints, while it does away witli the neces- 
sity of using the glue too thick, which should always be avoided ; 
for the less glue there is in contact with the joints, provided they 
touch, the better ; and when the glue is thick, it chills quickly, and 
cannot be well rubbed out from between the joints. The method 
to which we refer is, to rub the joints on Wxq edge with a piece 
(»f soft chalk, and, wiping it so as to take otf any lumps, glue it 
in the usual manner; and it will be found, Avlien the wood is porous, 
to Jiold much faster than if used without chalking. 

Of the differ.ent Methods of jolnhig Woodioorh. 

Many workrnen are not aware of the proportion which a piece 
made tQ. fit into, anotlu;;- shor.ld have towards that into which it is 



324 



Woodwork, Carpentry, etc. 



fitted, so as to produce the greatest strength with the least possible 
waste of material ; or how to proportion a joint, so that it shall not 
fail or give way before another. In too many instances, the 
method of joining woodwork is regulated by no other rule than the 
fancy of the workman. It is not difficult to explain why joiners' 
work so frequently fails; wliy the parts separate with a trifling 
strain ; or, from being bound too tightly together, fly and split in 
all directions. It is not so frequently from the bad execution of the 
work, as from the want of an adequate estimate of the strength 
required to resist the stress on the joint. We shall, then, describe 
the several kinds of joints, or the methods of framing and joining 
timber; and, under each head, give such directions, founded on the 
principles of mechanics, as will enable the workman to proceed 
with some degree of certainty ; and not, as is too frequently the 
case with artisans, observe no other rules than those which custom 
has authorized, and practice made familiar. 

Dovetailing. 

We have given, in the cuts, several examples of dovetailing. The 
parts which fit into each other are known by different names ; the 
projecting piece, represented in fig. 20, is called the pin of the 
dovetail; and the aperture into which it is fitted, as shown in fig. 





Fig. 20. Fig. 21. 

21, is called the socket. K'ow the strength of a dovetail depends 
upon so proportioning the pin and the socket as to enable them to 
support, rather than destroy, each other. Let A B C I), fig. 20, be 
a scantling, which is required to be joined to another, by means of 
a single dovetail. The strength of the joint depends on the form 
of the dovetail, as well as on the proportion it bears to the parts 
cut away. We shall endeavor to lay down the principle on which 
the greatest strength may be secured. Having squared the end of: 
the scantling, and gauged it to the required thickness, AIKLM, 
divide IM into three equal parts, at K and L. Let KL be the 
small end of the dovetail, and make the angles IKGandMLH 
equal to about 75 and 80 degrees respectively; and make GE and 
II F parallel to A N and B (). Then cut away the parts A I K G E N, 
and B M L H F O, and having formed the socket to correspond, by 



Woodwork, Carpentry, etc. 



S25 



I 




Fig. 22. 



marking the form of the dovetail on tlie top of the piece A B C D, 

fig. 21, and cutting away accordingly, the pieces may be fitted 

together, as shown in fig. 22. It may be here observed, that the 

bevel of the dovetail, that is, the angle 

I K G, fig. 20, may be either more or less 

than has been mentioned, according to the 

texture of the wood. Hard, close-grained 

woods, not apt to rive or split, will admit of 

a greater bevel than those which are soft, or 

subject to split; thus the bevel of a dovetail 

in deal must be less than in hard oak, or in 

mahogany. It is a great fault to make a 

dovetail too beveling, for instead of adding 

to the strength of the joint, as some persons suppose, it weakens it; 

for provided the bevel is sufficient to prevent the possibility of 

pulling the pieces apart, the less the bevel that is given the better. 

It must have been observed, that there is a great difference between 

the dovetail made by t!ie cabinet-maker and by the joiner; the 

former has very little bevel, the latter very much ; the former looks 

neat, and is at the same time strong; while the latter, appearing 

ix) aim at strength, looks clumsy, and is at the same time much 

the weaker of the two. 

Fig. 23 represents the dovetail in common use for drawer-fronts. 
When it is required to hide the appearance of the joint in front, 
the board ABCD is cut with the pin, and AEFB with the 
socket. The pins in this sort of dovetail are in general from about 
three quarters of an inch to an inch apart, according to the size of 
the pieces to be joined. 





Fig. 23. 



Fig. 24. 



Fig.' 24 represents the pin part of a lap dovetail, which, when 
put together, shows only a joint, as if the pieces were rebated 
together, as shown in fig. 25. ABC D represents the pin, EFG H 
the socket, and when put together the line H G is only seen as a 
joint; and if the corner AB is rounded to the joint G H, it will 
appear as if only mitred together. This kind of dovetail is very 
useful for many purposes where neatness is required, such as in 
making boxes. 



28 



326 



Woodwork, Carpentry, etc. 



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Fig. 25. 



Fig. 20. 



. Fig. 26 represents a still neater dovetail ; and, as the edges are 
mitred together, it is termed a mitred dovetail; and is the same as 
that sliown in fig. 6, except that instead of the square shoulder, or 
rebat, in A B, it is cut into a mitre, and the other piece is made to 
correspond. j 

Another very neat as Avell as expeditious method of joining pieces' 
of wood, and it is somewhat analogous to dovetailing, is shown in ' 
fig. 27. The joint is first formed into a mitre, and the pieces are 
then keyed together, either by making a saw kerf in a slanting 
direction, as at A B, or by cutting out a piece, as at C D, in the 
form of a dovetail. The first method, A B, is called, amongst work- 
men, keying together ; the second, C D, key-dovetailing. 




vx" 



Fig. 28. 

Th^e last method to be mentioned is that shown in fig. 28, and 
may be termed mitre dovetail grooving ; the part A B being formed 
with shoulders cut to the re(|^uired bevel, and a piece left for the 
pin dovetail, which is inserted into the socket dovetail, made to 
correspond to it in the pi^ce CD, which has been previously 
, formed into a mitre. This method, though not much employed, 
may be used with great advantage in many instances, particularly 
when it is required to join pieCves together the lengthway of the 
grain. ' » 

Mortice and Tenon, 

Under tills head, we shall end^aA'or to giv« «ome rules necessary 
to be observed in attempting to proportion the parts of the mortice 
and tenon, so that they may be equally etr^^ng, oi* that the tenon 
may not be m-ore likely to give way than the checks of the mor- 
tice ; for this is the principal thing to be avoided. The workman 



Woodwork, Carpentry, etc. 



327 



frequently allows too little substance for the tenon, lest he should 
weaken the moi'tice; and sometimes he falls into the opposite 
error; facts which clearly prove that he is not acquainted with a 
means of obtaining a maximum of strength with a given quantity 
of material. 

Figs. 29 and 30 represent a simple mortice and tenon. The dotted 




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Fig. 29. Fig. 30. 

lines show the parts to be cut awa}^ To show the thickness of the ; 
tenon, and consequently the Avidth of the mortice, we have here | 
one tenon and two shoulders, that is, three parts ; one of which is I 
to be allowed for the tenon, and two for the shoulders ; and this I 
will in general be found the best proportion, for if the tenon be i 
more than that, it will weaken the shoulders of the mortice. Now 
if we have, as is frequently the case, two tenons in one piece, as 
represented in fig. 31, there will be five parts, two tenons, and 
three shoulders; so that each tenon will be one fifth of the thick- 
ness of the stuff, for the shoulders are 
all equal to the tenons. This rule may 
be generally observed, unless the tenon 
is at a considerable distance from the 
end of the stuff, and then something 
more may be allowed for its thickness, 
as the mortice is then not so liable to 
split ; but it should in no case, how- 
ever sound the timber, or tough the 

material, be more than two out of four parts; that is to say, it can 
never be safe to make the tenon more than half the thickness of 
the stuff, and that only under particular circumstances, when the 
mortice is near the middle of the scantling, for the piece in which 
the mortice is cut would, in other cases, be considerably weakened. 
There is frequently in joiners' work a shoulder at the bottom of 
the tenon that fits into the piece 
in which the mortice is cut, as re- 
presented in fig. 32 ; and the tenon 
is divided into two parts, as there 
shown, which, when the stuff is 
wide, is a good method, as it 
strengthens the piece iji which the 
mortice is cut, witliout weakening, 
in the sarne proportion, the mortice 
itself; and we would suggest, in 
this case, tliat the piece 15 C, cut 
out from between the tenons A B 



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Woodwork, Carpentry, etc. 



I and D C, be nearl}*, if not quite, one tliird of tlie distance AD ; for 
! if much less, the piece left between the mortice will add but very 
' little to the strcDgth of the piece in Tvhich the mortice is made ; 
and the tenon would be stronger in proportion to the mortice-piece 
than necessary. It may be here observed, that if the width of the 
t-enon be much more than four times its thickness, additional 
strength will be gained by dividing the tenons into two or more 
parts, as shown in the figure, particularly^ if we allow a small piece 
at the bottom of the tenon, as represented in the drawing. 

! 

Grooving and Lapping, 

This method of joining wood-work is analogous to that of mor- 
1 ticing and tenoning. When it is required to join two boards toge- 
ther by means of a tongue and groove, the groove should never 
exceed one third of the thickness ; and often, if the piece for the 
tongue be formed of hard wood and liable to split, one quarter of 
the thickness will be sufficient. When a panel is let into a groove 
in the style, the joiner is often guided by the thickness of the panel 
itself, which should never be less than one third the thickness of 
the style. 

In making a groove across the grain, as for partitions, it will be 
best, in most cases, to make it about a fifth or sixth of the substance 
of the stuff. But, if the groove be formed into a dovetail, one 
quarter the thickness ^vill be better, and the dovetail should be i 
made a little tapering, but not too much. It should, in fact, be so 
made as to go almost home without requiring a blow from a 
hammer or mallet to drive it into its place until it has nearly 
attained it; and all joints should be easily separated with a gentle 
blow before they are glued. In a lap-joint, that is, in lapping two 
pieces together, supposing them of equal thickness, half the sub- 
stance of each should be cut away ; and, if of unequal thickness, 
the lap should be made in the thinner piece, of about two thirds or 
three quarters of its thickness, according to the substance of the 
thicker piece; thus endeavoring in this, as in all other cases, to 
avoid weakening one piece more than another. 

Bending and Ghdng-up. 

In bending and gluing-up stuff for sweep-work, much judgment 
is necessary, and, as the methods are various, we shall mention a 
few which the workman may apply, as occasion may require, one 
method being preferable to another, according to the nature of the 
work in hand. 

The first and most simple method is that of saw^ing kerfs or 
notches on one side of the board, thereby giving it liberty to bend 
in that direction ; but this method, though very ready and useful 
for many purposes, weakens the work, and may cause it to break 
when strains are thrown on the piece. But a tolerably strong 
sweep may be made in this manner, if, after sawing the kerfs 



Woodwork, Carpentry, etc. 329 



(particular care being taken to make them regular and even, and 
to saw them at regular depths), some strong glue be rubbed into 
each kerf. When bent into the required sweep, a piece of strong 
canvas should be glued over the kerfs themselves, and the glue be 
left to harden in the position to which the stuff is bent. | 

Another method is to glue up the stuff in thin thicknesses, in a | 
cawl or mould, made with two pieces of thick wood cut into the ! 
required sweep. This method, if done with care, that is, making! 
the several pieces of equal thickness throughout, of wood free from] 
knots, is perhaps the best that can be devised for strength and ; 
ae3uracy. It is also a practice sometimes to glue up a sweep in: 
three thicknesses, making: the middle piece the contrary way of the ; 
grain to the outside and inside pieces, which run lengthwise. This' 
method, tho:igh frequently used for expedition, is much inferior toi 
the above, as the different pieces cannot shrink together, and con- 
sequently the joint between them is apt to give way. | 

A solid piece, if not too thick, may be sometimes bent into thej 
form required. If a piece of timber be w^ell soaked upon the! 
intended outside of the curve, it may be bent into position, and if j 
kept in that position till cold will retain the curvature that is given' 
to it. I 

Tiie only other method of forming a curve, necessary for us to' 
mention, is that of cutting out solid pieces to the required sweep, | 
and gluing them upon one another till they liave the thickness i 
required, taking care that the joints are alternateljnn the centre ofj 
each pie3e below it, something in the manner of courses of bricks | 
one above the other. In this case, it will be necessary, if the work ; 
be not painted, to veneer the Avhole with a thin piece, after it has! 
been thoroughly dried and planed level, and then made somewhat 
rough with either a rasp or toothing-plane. But the joiner must 
adopt one plan or another, according to circumstances. 

Rcrihing. 
Scribing is the operation by which a piece of wood-work is made 
to fit against an irregular surface. Thus, for instance, the plinth 
of a room is made to meet or correspond with the unevenness of 
the floor. To determine the ]")ortion which is to be cut off from a 
partition, or any wood- work where a floor or ceiling is irregular, it 
is only necessary to open the compasses to a width equal to the 
greatest distance between the plinth and the floor; and, passing 
one leg over the uneven surface, the other leg will leave a mark on j 
the plinth. If the wood be cut away on that line, a surface will be 
obtained which will make a good joint with the floor or ceiling.' 
But the chief use of the art of scribing is to enable the joiner so to| 
connect the moulding of j^anels or cornices, that when placed' 
together, they shall seem to form a regular mitre-joint. This 
method lias certainly one advantage over the conunon method of 
mitring, for, if the stuff should shrink, little or no alteration will be 
made in the appearance, but, under the same circumstances, a mitre 

28* ■ 



330 Woodwork, Carpentry, etc. 



would open, and the joint would be shown. The method adopted 
is this : To cut one piece of the moulding to the required mitre, 
and then, instead of cutting the other to correspond with it, cut 
away the parts of the first piece to the edge of the first moulding, 
which will then fit to the other moulding, and appear as a regular 
mitre. 

Finishing of Joiners^ Work. 

Joiners' work is generally intended to increase the beauty of a 
building. When a joiner works in wainscot, oak, or mahogany, 
his chief object must be to obtain a surface perfectly smooth and 
even. When the framing is glued together, the glue which oozes 
out, and may be spilt upon the work, must be allowed to remain a 
few minutes and chill, and may then b6 carefully scraped off with 
a chisel; and the parts which cannot be thus cleaned may be 
washed with a sponge dipped in hot Witter and squeezed nearly 
dry. This not only saves trouble in operations which follow, but 
prevents staining, always produced when glue is suffered to remain 
till quite hard, particularly on wainscot, which turns black in 
every joint or place where the glue is suffered to remain. After 
this operation, which, though it may appear tedious to some work- 
men, will be found a saving of time, the work should remain till 
perfectly dry; and, when the joints and other parts have been 
levelled with a smoothing plane, the whole surface may be passed 
under a smooth scraper, and finished with fine glass-paper. It will 
be sometimes necessary, when the grain is particularly cross, to 
damp the entire surface with a sponge " to raise the grain," and 
then again to apply the glass-paper. The work will then be ready 
for polishing with wax, or varnishing, and the good appearance of 
the work will be in proportion to the time and trouble expended in 
the process. 

In cleaning pine, the same precautions must be taken for the 
removal of glue left upon the joints, or spilt upon the work, as 
already described. This being done, the work may be cleaned off 
with a piece of glass-paper that has been rubbed with chalk, or, in 
some cases, with a piece of hearthstone. The work is then ready 
for the painter ; but as there are knots and other places where the 
turpentine contained in the wood is apt to ooze out, either with or 
without the increase of heat, and thus spoil the appearance of the 
finishing, those parts are done over with a composition, and the 
process is called priming. This is properly the painter's business; 
but it must sometimes be done by the joiner, for the sake of saving 
his work. The composition used for this purpose is made with red 
lead, size, and turpentine, to which is sometimes added a small 
quantity of linseed oil. Priming has also the advantage of pre- 
venting the knots from being seen through the paint. Some work- 
men omit in this composition the oil and the turpentine, but the 
size of itself is apt to i)eel off, and does not thoroughly unite itself 
with the wood. 



Terms used in Building. 



331 



Another method of cleaning-off pine is sometimes adopted. "When 
the surface has been made qnite smooth with the plane, it is rubbed 
with a pieee of chalk, and the whole is cleaned with a piece of fine 
pumice-stone, as in the former process it was done with glass- 
paper ; but if the grain should be still rough, the work may be 
damped with a sponge, and the operation repeated when dry. 

As, in finishing interior work, it is now^ customary to imitate the 
graining of different kinds of wood, it is necessary that the joiners' 
work should be well finished ; for if a good even surface be not 
provided, it will be impossible for the painter to produce the effect 
he desires. Every defect in the ground will, in fact, be more 
visible under a delicate graining than when the surface is covered 
with successive coats of color ; but, even in the latter case, work 
well prepared will not only look better, but the color will not be 
so apt to chip and peel off as when the surface is not properly 
levelled. 



TERMS USED IN BUILDING. 

Abacus. — The upper member of the capital of a column, that on 
which the architrave rests. It has different forms in the several 
orders : In the Tuscan or Doric, it is a square tablet ; in the Ionic, 
its edges are moulded ; in the Corinthian, its sides are concave, and 
frequently enriched with carving. 

Ahitmeut. — That part of a pier from which the arch springs. 

Acanthus. — A plant whose leaves arc carved on the Corinthian 
and Composite capital. They are differently disposed, according 
to circumstances ; and the leaves of the laurel and parsley are 
sometimes employed in their place. 

Acroterium. — A pedestal on the angle or apex of a pediment, 
intended as a base for sculpture. 

Altitude. — The perpendicular height of anything in the direction 
of the plumb line. The length of a body is measured on the body 
itself, and remains constant, its altitude varies according to its 
inclination to or from the perpendicular. 

Alto Relievo. — A sculpture, the ligures of which project from the 
surface on which they are carved. 

Amphiprostylos — An order of Grecian temples, having columns 
in the back as well as the front. 

Amphitheatre. — A double theatre, employed by the ancients for 
public amusements. The colosseum at Eome, built by Yespasian, 
is one of these. 

Annulet. — A small square moulding, used to separate others; 
the fillet wliich separates the flutings of n column is sometimes 
known by this term. 



332 Terms used in Blilding. 



A7itce. — Pilasters attached to a wall, receiving an entablature, 
and having bases and capitals differing according to the order 
employed, but always unlike those of the columns. 

Antepagmejita. — A term in ancient architecture, the architraves 
round doors 

Apophyge. — That part of a column which connects the upper fillet 
of the base and the under one of the capital with the cylindrical ; 
part of the sliaft. | 

Arceosti/los — That style of building in which the columns are' 
distant from one another from four to five diameters. Strictl}^ j 
speaking, the term should be limited to an intercolumniation of; 
four diameters, which is onl}^ suited to the Tuscan order. i 

Arch. — Such an arrangement, in a concave form, of building! 
materials, as enables them, supported by piers or abutments, to! 
carry weights and resist strains. | 

Arch-buttress. — Sometimes called a flying buttress; an arch 
springing from a buttress or pier against a wall. 

Arckltrave. — That part of the entablature which rests upon the 
capital of a column, and is beneath the frieze. It is supposed to 
represent the principal beam of a timber building. 

Area. — This term is applied to superficies, whether of timber, 
stone, or other material, and is the superficial measurement ; that 
is, the length multiplied into the breadth. The word area some- 
times signifies an open space. 

Arris — The line in which two surfaces meet each other. 

Ashler. — Common freestone, as it comes from the quarry, gene- 
rally about nine inches thick, but of difi"erent superficial dimen- 
sions. 

Aahlerhig. — Quartering, to which laths are nailed. 

Astragal. — A small moulding Avith a semicircular profile, some- 
times plain and sometimes ornamented. 

Attk Order. — A term used to denote the low pilasters which are 
placed over orders of columns or pilasters, and frequently employed 
in the decorations of an attic. 



B. 

Baluster. — A small pillar or pilaster, supporting a rail. 

Balustrade. — A series of balusters connected by a rail. 

Baud. — A square member. To distinguish the situation in which 
it is placed, or the order in which it is used, an adjective is fre- 
quently prefixed ; thus, a dentil or a modillion band. 

Base. — The lower division of a column. The Grecian Doric has 
no base, and the Tuscan has only a single torus on a plinth. 

Bead. — A circular moulding, which lies level with the surface of 
the material in which it is formed. When the moulding projects, 
or several are joined, it is called reeding. 

Beak. — A small fillet in the under edge of a projecting cornice, 



Terms used in Building. 



intended to prevent the rain from passing between the cornice and 
fascia. 

Beam.— A piece of timber in a building laid horizontally^ and 
intended to support a weight, or to resist a strain. 

Beam-filling. — The masonry, or brickwork, between beams or 
joists. 

Bearer. — A vertical support. 

Bearing. — The length between bearers, or walls ; thus, if a beam 
rests on walls twenty feet apart, the bearing is said to be twenty 
feet. 

Bed Mouldings. — Those mouldings between the corona and the 
frieze. 

Bevil. — An instrument used by workmen for taking angles. In 
form it resembles a square, but the blade is moveable about a 
centre. When the two sides of any solid body have such an incli- 
nation to each other as to form an angle greater or less than a 
right angle, the body is said to be beviled. ^ 

Bond. — A term used to signify the connection between the parts 
of a piece of workmanship. In brickla3'ing and masonry, it is that 
connection between bricks, or pieces of stone, which prevents one 
part of the building from separating itself from another. 

Bond Timber. — Timber laid in walls to tie or bind them together. 

Brace. — A piece of timber placed in an inclined position, and 
used in partitions or roofs, to strengthen the framing. When a 
brace is employed to support a rafter, it is called a strut. 

Bressummer. — A beam, or iron tie, intended to cany an external 
wall, and itself supported by piers or posts. 

Bricknoggin. — Brickwork between quartering. 

Buttress. — A mass of stone or brick-work intended to support a 
wall, or to assist it in sustaining the strain that may be upon it. 
Buttresses in Gothic architecture are used for ornament as well as 
strength. 

C. 

Cabling. — Cylindrical pieces filling up the lower part of the flutes 
of a column. 

Camber. — To give a convexity to the upper surface of a beam. 

Cantalivers. — Pieces of wood or stone beneath the eaves to sup- 
port them, or mouldings above them. 

Capital. — That part of a colunm or pilaster beneath the enta- 
blature ; or, in other words, the uppermost member of a column or 
pilaster. The capital is variously formed, according to the order: 
Thus, we have the Tuscan, Doric, Ionic, Corinthian, and Composite 
capitals, and many others, that have been invented since the times 
of the Greeks au'l liomans. 

Caryatides. — Figures of women, introduced to support an enta- 
blature, instead of columns. 



334 Terms used in Building. 



Casement. — Applied to a window which is hung upon hinges in 
place of lines and weiglits. 

Castmg. — Tiie warping or shrinking of timber or wood-work, 
occasioned by an insufficient strengtli, or by an unequal exposure 
to the weather, and want of proper seasoning. 

Caoetto. — A concave moulding, the quadrant of a circle. 

Centering. — Tlie framing upon which an arch is turned. 

Clamping. — AVhen one piece of wood is so fixed into the end of 
another as to prevent it from splitting or casting, it is said to be 
clamped. The pieces may be united with a mortice and tenon, or 
with a groove and tongue. 

Collar Beam. — A beam framed between two principal rafters. 

Console. — An ornament cut on the key-stone of an arch, some- 
times in the form of a scroll, at other times to represent a human 
face. 

Content.' — The amount of any substance in rods, yards, feet, or 
inches whether solid or superficial. 

Coping. — The stone which covers the top of a wall or parapet. 

Corbel. — K bracket, or piece of timber projecting from a wall : in 
Gothic architecture, usually carved with some grotesque figure. 

Cornice. — The combination of mouldings which finishes or crowns 
an entablature. — The term is also applied to the mouldings which 
finish the walls and ceiling of a room, hall, or passage, filling up 
the angle which they make. 

Crown. — A term applied to the uppermost or highest part of an 
arch, that in which the key-stone is fixed. 

Cyma. — A moulding with a waved or crooked profile, partly 
convex, partly concave. It is called b}' workmen an ogee. When 
the hollow part of the moulding is uppermost, it is called a cynia- 
recta ; when the convex part is above, a cyma-reversa. 



D. 

Dado. — That flat part of the base of a column between the plinth 
and the cornice. It is of a cubical form, and from thence takes its 
name. 

.Dentils. — Square blocks introduced as ornaments into cornices of 
the Doric, Ionic, and Corinthian orders. A small circular piece is 
sometimes cut out, and at other times they are fluted. 

Die. — A square cube. 

Door Frame. — The case in which a door opens and shitts, consist- 
ing of two uprights and one horizontal piece, connected together 
by mortices and tenons. 

Dormer — A window made in the sloping part of a roof, or above 
the entablature. 

Dovetailed — When two pieces of wood are fastened together, by 
letting the pieces of one into apertures formed in the other, of a 



1 



Terms tsED in Building. 335 



feliape somewhat resembling a fan or dovetail, they are said to be 
dovetailed. 

i>r(?/?.s— Ornaments in the Doric entablature resembling bells 
placed immediately under the triglyphs. 

Dwarf Wall. — A wall that has a less height than that of the 
story in which it is used, 

E. 

leaves. — The edge of a roof or slating which overhangs a wall, 
and is designed to carry off the water, without flowing down the 
wall. 

jEJchinus. — A moulding, the profile of which is the quadrant of a 
circle turned outwards, or in some instances a conic section. It is 
said to resemble the shell of the chestnut. 

Mlipse.^-That curve called by workmen an oval. 

Entablature. — That assemblage of mouldings, (fee, which are sup- 
ported by the column. It consists of three parts — the architrave, 
frieze, and cornice. 

Entasis. — The swelling of a column. 

Eustijlos — That intercolumniation in which the columns are 
placed two diameters and a quarter from each other. 

Eye. — A term sometimes used, in architecture to denote a small 
window in a pediment. The middle of the Ionic volute, that is, 
the circle within which the different centres for drawing it are 
found, is known by the same name. 

F. 

Facade. — The face or front of a building ; strictly speaking, the 
principal front. 

taF^ascia. — A flat broad member, in architecture, but of small pro- 
jection. It is used to denote the flat members into which the 
architrave is divided, and these are called fascia?. 

Feather-edged. — Boards or planks thicker at one edge than the 
other. 

Fillet. — A small square moulding, of slight projection. In car- 
pentry, it means a piece of wood to which boards are nailed. 

Flashings. — Pieces of lead so let into the wall as to lap over a 
gutter. 

Flatting. — Painting, which has no gloss on its surface, being 
worked with turpentine. It is used for finishing. 

Flutes. — Vertical channels cut in the shafts of columns and pilas- 
ters, sometimes meeting one another at a sharp edge, and at other 
times having a fillet between them. 

Flyers. — Stairs which rise without winding. 

Flue. — The aperture of a chimney. 

Footings. — The courses of brick or stone at the foundation of a 
wall. 



336 Terms used in Building. 



Frieze. — The flat member in an entablature, separating the archi- 
trave from the cornice. 

Furring. — A means of restoring an irregular framing by the 
addition of small pieces of wood nailed to the framing itself. 

Fust. — The shaft of a column. 



G. 

Gable. — The upright triangular end of a building at the ends of a 
roof. 

Girder. — The largest piece of timber in a floor, that into which 
the joists are framed. 

Groin. — The intersection of two arches. 

Groove. — A rectangular channel cut in stone or timber ; such as 
that which is cut in the stiles to receive the panel of a door. 

Grounds. — Those pieces of wood imbedded in the plastering of 
walls to which skirting and other joiners' finishings are attached. 

Guttce.—See "Drops." 

Gutter. — A valley between the parts of a roof, or between the 
roof and parapet, intended to carry off the rain. 

n. 

Half Hound. — A moulding in a semicircular form, projecting from 
the surface. 

Headers. — Bricks laid with their short face in front. 

Hips. — Those pieces of timber placed in an inclined position at 
the corners or angles of a roof. 

I. « 

Impost. — The combination of mouldings which form the eaipital 
of a pier. 

Insidated. — A term applied to a column Avhich is unconnected 
with a wall, or to a building, that stands detached from others. 

Inter colwnniati on. — The space between two columns. 

Intertie. — Small pieces of timber placed horizontally between, 
and framed into, vertical pieces to tie them together, 

J. 

Jambs — The side pieces of an opening in a wall, such as door- 
posts, and the uprights at the side of window frames. 

Joggle-piece. — A post to receive struts. 

Joists. — Those pieces of timber which are framed into a girder, 
bressummer. or otherwise, to support a ceiling or a floor. 



Terms used in Building. SS7 



Key-stone. — That stone in the top or crown of an arch which is 
I in a perpendicular line with the centre. i 

j King-post. — The centre post of a trussed framing, intended to 
support the tie-beam and struts. 

Knee. — A piece of timber bent to receive some weight, or to 
I'd i eve a strain. 

! . . . .' 

I Lantern. — A frame in the dome or cupola of a building to give 
i light. The term is applied to some kinds of fanlights, that is, the' 
frame over a door to light a passage or corridor. 

Lining. — That joiners' work which covers an interior surface. 

Lintels. — The pieces of timber which lie horizontally over the 
jambs of windows and doors. - 

M. 



Mantel. — The cross-piece which rests on the jamb of a chimney. 

Metopa. — The interval between the triglyphs in the Doric order. 

Minute. — -The sixtieth part of the diameter of a column. j 

Modillion. — An ornament in the Ionic, Corinthian, and Composite ! 
orders. It is a sort of bracket, and sliould be placed under the! 
corona. I 

Module. — The semi-diameter of a column, and is divided into ■ 
thirty minutes. It is the measure by wliich the architect deter- i 
mines the proportions between the parts of an order. ! 

Mortise. — A method of joining two pieces of wood; a hole being; 
made in one of such a size as to receive the tenon or projecting; 
piece formed on the other. i 

Mosaic. — A term applied to pavements, and other work, when , 
formed of various materials of different shapes and colors, laid in a ! 
kind of stucco, so as to present some pattern or device. The- 
ancients were very successful in the execution of Mosaic, and many 
fine specimens remain to this day. 

Midlion — Upright posts or bars which divide the lights in a 
Gothic Avindow. 

N. 

Naked. — This term is applied, in architecture, to a plain surface, 
or that which is unfinished ; as the naked walls, the naked flooring 
— that is, uncovered. The word is sometimes applied to flat sur- 
faces before the mouldings and other ornaments have been fixed. 

Newel. — The centre round which the stairs Avind in a circular 
staircase. 



29 



J^38 Terms used in Building. 



Nosings. — The rounded and projecting edges of the treads of 
stairs. 

O. 

Obelisk. — A slender pj'ramid. 

Oc/ee. — A moulding, consisting of a portion of two circles turned ' 
in contrary directions, so that it is partly concave and partly con- 1 
vex, and somewhat resembles the letter S. I 

Order. — An assemblage of parts having certain proportions to| 
one another. There are five orders of architecture — Tuscan, Doric, \ 
Ionic, Corinthian, and Composite — all of which were invented by| 
the ancients, and are now employed by the moderns. ; 

Oval. — A curve line, the two diameters of whicli are of unequal 
length, and is allied in form to the ellipse. An ellipse is that figure 
which is produced by cutting a cone or cylinder in a direction 
oblique to its axis, and passing through its sides. An oval may be 
formed by joining different segments of circles, so that their meet- 
ing shall not be perceived, but form a continuous curve line. All 
ellipses are ovals, but all ovals are not ellipses ; for the term oval 
may be applied to all egg-shaped figures, those which are broader 
at one end than the other, as well as to those whose ends are 
equally curved. 

Ovolo. — A convex projecting moulding whose profile is the qua- 
drant of a circle. 



Panel. — A compartment inclosed in a frame, into which it is 
framed or grooved. 

Parapet. — A low wall generally about breast-high, on the top of 
bridges or buildings. 

Pargetting. — Eough plastering, commonly adopted for the inte- 
rior surface of chimneys. 

Pedestal. — That arrangement on which columns are sometimes 
placed : it is divided into three parts — the cornice, the die, and tlie 
base. 

Pedhnent. — A low triangular crowning ornament in the front of 
a building, and over doors and windows. Pediments are sometimes 
made in the form of a segment of a circle. 

Pier. — A square, or other formed mass, used to strengthen or 
support a building; it sometinies signifies that mass of stone or 
brickwork between the arches of a bridge, and from which they 
spring, or against which they abut But the term is usually 
employed to designate the solid part between the doors or windows 
of a building. 

Pilaster. — A square pillar insulated, or engaged to the wall, and 
is usually enriched with a capital and base. 

Piles. — Large timbers, usually shod with pointed iron caps, 



Terms used in Building. 339 



driven mto the ground for the purpose of making a secure foun- 
dation. 

Fillar. — An irregular, insulated column. It differs from a column 
in having no architectural proportion, being either too massive or 
too slender. 

Pinnacle. — A small spire used to ornament Gothic buildings. 

Pitch of a Roof. — The proportion obtained by dividing the span 
by the height ; thus we speak of its being one half, one third, one 
fourth. 

Plinth — The solid support of a column or pedestak 

Plumb-line. — An instrument to determine perpendiculars ; it con- 
sists of a piece of lead attached to a string. 

Porch. — The vestibule or entrance to a building. 

Portico. — A kind of gallery or piazza, frequently erected in front 
of large buildings. 

Pos^.s.— Square timbers set on end ; the term is especially applied 
to those which support the corners of a building, and are then 
framed into the bressummer or cross-beam, under the walls. 

Pricking-up. — The first coat of plaster worked upon laths. 

Profile. — The outline; the contour of a part, or the parts com- 
passing an order. 

Pugging. — The stuff laid upon sound boarding to prevent the 
passage of sound from one story to another. 

Puncheons. — Short pieces of timber employed to support a weight 
when the bearing is too distant. 

Purlines. — Those pieces of timber which lie across the rafters to 
prevent them from sinking. 

Putlogs. — Pieces of timber used in building a scaffold ; they are 
those which lie at right angles to the line of wall, and rest on the 
scaffold poles or ledgers. 

Pyramid. — A solid massive edifice which rises from a square or 
rectangular base, and terminates in a point called the vertex. 

Q. 

Quarter Round. — See " Ovolo." 

Quarters. —Pieces of timber used in an upright position for par- 
titions. Quarters may be either single or double; the single are 
generally two inches thick, and four inches broad; the double are 
four inches square. The quarters are never placed at a greater 
distance than fourteen inches from each other. 

Quirk. — A piece of ground taken out of a plot. The term is also 
applied to a particular form of moulding, one which has a sudden 
convexit}'. 

Quoins. — The corners of a building; they are called rustic quoins 
when they project from the wall, and have their edges chamfered 
off. 



340 Terms used in Building. 



R. 

Rahhet or Rebate. — A groove or channel in the edge of a board. 

Rafters. — Those timbei^ -syhioh form the inclined sides of a roof. 

Raking. — Means literally iiicliiiiDg, and is applied to those mould- 
ings which, instead of maintaining the horizontal line, are suddenly 
bent out of their course. 

Rails. — Those pieces in framing which lie in a horizontal position 
are called rails; those which are perpendicular are called stiles; 
hence two rails and two stiles inclose a panel. The term is also 
applied to those pieces in fences or paling which go from post to 
post. 

Relief. — The projection which a figure has from the ground on 
which it is carved. 

Return. — That part of any work which falls away from the line 
in front. 

Ridge. — The highest part of a roof, or the tnnber against which 
the rafters pitch. 

Riser — That board in stairs set on edge under the tread or step 
of the stair. 

Rustic. — This term is applied to those courses of stone-work, the 
face of which is jagged or pecked so as to present a rough surface. 
That work also is called rustic in which horizontal and vertical 
channels are cut in the joinings of the stones, so that when placed 
together an angular channel is formed at each joint 

S. 

Sash. — The framework which holds the squares of glass in a 
window. 

Sash-frame. — ^The frame which receives the sash. 

Scantling. — The measure to which a material is to be or has been 
cut. 

Scotia. — A semicircular concave moulding, chiefly used between 
the tori in the base of a column. 

Scribing. — Fitting wood-work to an irregular surface. 

Scroll. — A carved curvilinear ornament, somewhat resembling in 
profile the turnings of a ram's horn. 

Sill. — The horizontal piece of timber at the bottom of framing 5^ 
the term is chiefly applied to those pieces of timber or stone at the 
bottom of doors or windows. 

Shaft. — The body of a column ; that part between the base and 
capital. 

Shore. — A piece of timber placed in an oblique direction to sup- 
port a building or wall. 

Skirting. — The narrow boards placed round an apartment against 
the walls, and standing vertically on the floor. 

Sleepers. — Pieces of timber placed on the ground to support the 
ground-joists, or other woodwork. 



Terms used in Bulldixg. 341 



Soffit. — A term applied to a frame or paneling overhead, or to a 
lining, such as that which is fixed in the underside of the tops of 
windows. 

Stiles. — The upright pieces in framing or paneling. 

Struts. — Pieces of timber which support the rafters. 

Summer. — A large piece of timber supported by piers or posts; 
when it supports a wall, it is called a breast-summer, or bres- 
summer. 

T. 

Tenon. — A piece of wood so formed as to be received into a hole 
in another piece called a mortice. 

Throat. — That hollow which terminates the upper end of the 
shaft of a column. 

Tongue. — That projecting piece at the end of a board which is 
formed to be inserted into a groove. _ 

Toru^. — A moulding that has a convex semicircular or semi- 
elliptical profile. 

Transom — A piece that is framed across a double window-light. 

Trellis. — An open framing, pieces crossing each other so as to form 
diamond or lozenge-shaped openings. 

Tryglijphs. — Ornaments in the Doric frieze consisting of a square 
projection with two angular channels, the edges of each forming 
half a channel. They are placed immediately over the centre of a 
column ; their width is generally one module. 

Trimmers. — Pieces of timber framed at right angles to the joist 
for chimneys, and the av ell-holes of stairs. 

Tympanum. — The space inclosed by the inclined and horizontal 
sides of a pediment. 

V. 

Valley — The space between two inclined sides of a roof. 

Vaults. — Underground buildings with arched ceihngs, whether 
circular or elliptical. 

Vertex. — The top or summit of a pointed body, as of a cone. 

Volute. — The scroll in the capital of the Ionic order. 

Voussoirs. — The stones which compose the face of an arch, 
having a somewhat wedge-shaped form. 

W. 

Wall-plates. — The timbers built up with a wall, to carry the 
joists. 

Weather-boarding. — Weather-edge boards, fixed vertically, so as 
to lap over one another. 

Well-hole. — The aperture left in floors to bring up the stairs. 



29* 



342 Glues — Papers. 

GLUES. 

Parchment Glue, 
Parchraent shavings 1 pound ; water 6 quarts. Boil until dis- 
solved, then strain and evaporate slowly to the proper consistence. 
Use a water bath if you want it very light colored. 

Japaiiese Cement^ or Rice Glue. 
Rice flour; water, sufficient quantity. Mix together cold, then 
bring the mixture to a boil, stirring it all the time. Observe to 
boil it in a vessel that will not color it. 

Japanners' Gold Size. 
Gum ammoniac 1 pound ; boiled oil 8 ounces ; spirits of turpen- 
tine 12 ounces. Melt the gum, then add the oil, and lastly the 
spirits of turpentine. 

Gold Size. 
Yellow ochre 1 part ; copal varnish 2 parts ; linseed oil 3 parts ; 
turpentine 4 parts ; boiled oil 5 parts. Mix. The ochre must be in 
the state of the finest powder, and ground with a little of the oil 
before mixing. 

Glue Liquid. 
Glue, water, vinegar, each 2 parts. Dissolve in a water-bath, 
then add alcohol 1 part. An excellent cement. 

Transparent Liquid Japan for Metal. 
Copal varnish 35 parts ; camphor 1 part; boiled oil 2 parts. Mix. 

Portable Glue for Draughtsmen, d:c. 

Glue 5 parts ; sugar 2 parts ; water 8 parts. Melt in a water- 
bath, and cast il in moulds. For use, dissolve in warm water. 

Waterproof Glue. 

1. Glue 1 part ; skimmed milk 8 parts. Melt and evaporate in a 
water-bath to the consistence of strong glue. 

2. Glue 1 2 parts ; water sufficient to dissolve. Then add yellow 
resin 3 parts, and when melted add turpentine 4 parts. Mix 
thoroughly together. This should be done in a water-bath. 



P APEES. 

Fire-proof Paper. 
Take a solution of alum and dip the paper into it, then throw it 
over a line to dry. This is suitable to all sorts of paper, whether 
plain or colored, as well as textile fabrics. You must try a slip of 
the paper in the flame of a candle, and if not sufficiently prepared 
dip and try it a second time. 



Papers. 343 

Black Edge Paper. 
Blacklead 11 parts; common ink 22 parts; dissolved gum-arabic 
1 part. Mix. Then with a sponge lay the color on the edge of the 
I^aper, previously placed in the cutting-press, rub it in with a piece 
of cloth, and burnish. The edge of the paper must be rendered 
perfectly smooth before applying the black. 

To Stain Paper or Parchment. 
Red. — Brazil 12 parts ; water '70 parts ; alum 5 parts. Boil. 

1. Blue. — Sulphate of indigo. Water to dilute. 

2. Prussian blue 2 parts ; muriatic acid 1 part. Water to dilute. 

3. Logwood 4 parts ; water 30 parts ; sulphate of copper 1 part. 
Mix. 

Green. — Crystals of verdigris 2 parts; vinegar 1 part. Water to 
dilute. 

Yellow. — French berries, water, and a little alum. Boil. 

Purple. — Logwood 2 parts; alum 1 part;^vater 20 parts. Boil. 
The addition of a little gum to the above renders them suitable for 
coloring maps, <fec. 

Paper for Draughtsmen^ 6:c. 
Powdered tragacanth 1 part; w^ater 10 parts. Dissolve and 
strain through clean gauze, then lay it smoothly with a painter's 
brush on the paper, previously stretched on a board. This paper 
will take either oil or water colors. 

Copying Paper. 
Lay open 3'our quire of paper (clean white, of large size), take 
the brush and cover it with the following varnish, then hang it up 
on the line ; take another sheet and repeat the operation, until you 
have finished your quantity. If not clear enough, give each sheet 
another coat when dry : — Canada balsam, turpentine, equal parts. 
Mix. 

Liquid Gold^ for Vellum^ dec. 
Take gold-leaf and grind it with gum-water ; then add a small 
quantity of bichloride of mercury, and bottle for use. 

Liquid Silver, for Vellum, &c. 
Take silver-leaf and grind it, with gum-water or glair of ^o^g. 

Paper that Resists Moisture. 
Take unsized paper, lay it flat on a clean surface, and brush it 
over with a solution of mastic in oil of turpentine ; or plunge it 
into the solution and hang it up to dry. This paper possesses all 
the usual qualities of writing paper, with the advantage of resisting 
moisture. 

To Detect the presence of Plaster in Paper. 
Calcine the paper in a close vessel, and dilute the residue with 



344 Papers. 

vinegar, in a silver spoon ; if sulphuretted hydrogen is disengaged, 
which blackens the spoon, the presence of a sulphate (plaster) will 
be shown. This adulteration has lately become very common 
among the paper-makers, with the view of increasing the weight. 

Waxed Paper. 

Take cartridge or other paper, place it on a hot iron and rub it 
with beeswax, or make a solution of the wax in turpentine, and 
apply it with a brush. Useful for making water or air-proof pipes, 
for chemical experiments, <fec. \ 

To extract Grease Spots from Paper. j 

Apply a little powdered pipe-clay, on which place a sheet of; 
paper, then use a hot iron. Remove the adhering powder with a' 
piece of India-rubber. 

Papier Mdche. 
Take paper, any quantity. Boil it well, then pound it to a paste, 
and mould. Used m making toys, snuff-boxes, &c. 

To Gild the Edges of Paper. 
Armenian bole 4 parts ; sugar candy 1 part. White of ^g^ to 
mix. Apply this composition to the edge of the leaves, previously 
firmly screwed in the cutting-press ; when nearly dry smooth the 
surface with the burnisher ; then take a damp sponge and pass 
over it, and with a piece of cotton-wool take the leaf from the 
cushion and apply it to the work ; when quite dry burnish, 
observing to place a piece of silver or India paper between the 
gold and the agate. 

Tracing Paper. 
IS'ut oil 4 parts ; turpentine 5 parts. Mix, and apply it to the 
paper, then rub it dry with wheat flour, and brush it over with ox- 
gall. This will bear writing on. 

Lithographic Paper. 
Give the paper 3 coats of thin size, 1 of starch, and 1 of solution 
of gamboge. Each to be applied with a sponge, and allowed to 
dry before the next is applied. 

Hydrographic Paper. 
This name has been given to paper which may be written on 
wdth water. It may be made by rubbing paper over with a mix- 
ture of finely powdered galls and sulphate of iron heated till it 
becomes white. The powder may be pressed into the paper by 
passing it between rollers, or passing a heavy iron over it. A mix- 
ture of dried sulphate of iron and ferro-prussiate of potash may be 
used for blue writing. Or the paper may be imbued with a strong 
solution of one ingredient thoroughly dried, and the other applied 
in powder. Paper which has been wet with a solution of ferro- 



Papers. 345 

prussiate of potash also serves for writing on with a colorless solu 
tion of persulphate of iron. 

Iridescent Paper. \ 

Nut-galls 8 parts ; sulphate of iron 5 ; sal-ammoniac 1 ; sulphate \ 
of indigo 1 ; gum-arabic \. To be boiled in water, and the paper i 
washed with it exposed to ammonia. 

To give Paper the Appearance and Toughness of Parchment 
Dip white unsized paper for half a minute in strong sulphuric 
acid, and afterwards in water containing a little ammonia. When 
dried it Avill look like, and be as strong as parchment. 

Photographic Papers. 
The following papers should be the finest satin post, of uniform 
texture, free from the maker's mark, specks, and all imperfections. 
The papers must be prepared by candle-light, and kept in the dark 
till used. ^ 

1. Simple Nitrated Paper. — This is merel}^ paper brushed over 
with a strong solution of nitrate of silver. In brushing over the 
paper it must be crossed. Its sensitiveness is increased by using 
spirit of wine instead of water. This paper only requires washing 
in water to fix the drawing. 

2. Muriated Paper. — The paper is first soaked in solution of 
copper salt, pressed with a linen cloth or blotting paper, and dried. 
It IS then brushed over on one side (which should be marked near 
the edge) with the solution of nitrate of silver, and dried at the 
fire. The stronger the solution the more sensitive the paper. If 
dipped in a solution consisting of 35 grains of chloride of barium 
and 2 oz. of distilled water, richer shades of color are obtained. 

3. Iodized Paper. — Brush over the paper on one side (which 
should be marked) with strong solution of nitrate of silver (100 gr. 
to 1 oz ) ; tlien dip it in a solution consisting of 100 gr. of iodide of 
potassium dissolved in 4 oz. of distilled water. Wash it in dis- 
tilled watei', drain, and dry it. 

4. Bromide Paper. — Soak the paper in a solution composed of 
40 gr. bromide of potassium dissolved in 1 oz. of distilled water ; 
then brush it over with a strong solution of nitrate of silver, and 
dry in the dark. 

5. Calotype Paper. — Dissolve 100 gr. of crystallized nitrate of 
silver in 2 oz. of distilled water, and add 2 fluid dr. and 40 minims 
of acetic acid. Mix these at the time of using with an equal mea- 
sure of cold saturated recently prepared solution of gallic acid. 
Brush iodized paper with this solution, and mark the side ; in half 
a minute dip it into water, and press it between blotting paper. It 
is then ready for the camera, where it remains from lialf a minute 
to 5 minutes. When removed from the camera dip it into water, 



346 Papers. 

press it betAveen blotting paper, and wash it with a solution of 100 1 
gr. of bromide of j^otassium in 8 or 10 oz. of water. i 

6. Chromotype Paper. — Soak the paper in a solution of bichro-| 
mate of potash (in which solution a little sulphate of indigo is some- i 
times added to vary the color), and dry it at a brisk fire. To fix 
the drawing careful immersion in warm water is all that is required 
It is not sufficiently sensitive for the camera. 

'7. Compound Chromotype Paper. — Dissolve 10 gr. of bichromate ; 
of potash, and 20 gr. of sulphate of copper, in an ounce of water, j 
Wasli the paper in this solution, and dry it. After the paper has ; 
been exposed to the sun, with the article to be copied superposed . 
upon it, it is washed over in the dark witli a solution of nitrate of; 
silver of moderate strength, A vivid picture makes its appearance, 
which is sufficiently fixed by washing in pure watei*. This is foi' 
co])ying engravings, cfec. Another method is to brush writing 
paper over w^ith a solution of 1 dr. of sulphate of copper in 1 oz. of 
water; and when dry wuth a strong, but not saturated, solution of 
bichromate of potash. 

8. Cyanotype Paper. — Brush the paper over vvith a solution of 
ammonio-citrate of iron. Expose the paper in the usual way, then 
wash it over with a solution of ferro-cyanide of potassium. 

9. Crysotype Paper. — Wash the paper with solution of ammonio- 
citrate of iron, dry it, and afterwards brush it over with a solution 
of ferro-cyanide of potassium. Dry it in a dark room. The image 
is brought out by brushing it over with a neutral solution of gold 
or silver. 

10. Catallsotype. — Steep paper in water, with a drop or two of 
hydrochloric acid; absorb the superfluous moisture with blotting 
paper; brush over with a mixture of ^ dr. syrup of iodide of iron, 
2^ dr. of water, and a drop or two of tincture of iodine. Dry wi:h 
blotting paper, and brush over with a solution of 12 gr. of nitrate 
of silver to 1 oz. of distilled water. It is then read}^ for the camera. 
The picture is fixed by washing in water, and afterwards in a solu- 
tion of 20 gr. of bromide to 1 oz. of potassium. 

11. Paper for Positive Photographs. — Most of the preceding give 
negative pictures, the lights and shadows being reversed ; in the 
following they are correct: Dissolve 40 gr. of muriate of ammonia 
in 4 oz. of w^ater. Wash highly glazed paper in this solution, dry it, 
and brush it over with the followmg solution : Dissolve 120 gr. of 
crystallized nitrate of silver in 1^ oz. of distilled water; and add 
l^ oz. of alcohol ; after it has stood a few hours filter it. Expose ; 
the paper thus washed to the sunshine, till it is darkened ; if mot- 
tled, wash it a second time, and expose it again. Before using the 
paper make up the foUow^ing solution : H^'driodate of bar^'tes 40 j 
gr. ; water 1 oz. ; pure sulphate of iron 5 gr. Mix, filter, add a 
drop or two of diluted sulphuric acid, and when settled decant the 
clear liquor for use. Wash the paper over in this solution, expose 



Broxzing 347 

it in the damp «tate, with the engiaving or other object on it to the 
light, and fix the drawing by waslung with water only. 

Photographs. 

To copy objects, lay them on a plate of clear glass, fixed in a 
frame; place the prepared paper over them ; and fix a back, with a 
cushion attached to it, so as to press the paper closely on tlie glass. 
The glass is then exposed to the light, and the drawing afterwards 
fixed, as described above. For feathers, lace- work, and other 
objects which freely admit light through them, the nitrated paper 
and less sensitive muriated papers may be used. For copying 
engravings, leaves, and other botanical objects, or entomological 
specimens, the more sensitive muriated papers, or the bromide 
paper, or other sensitive kinds, may be used. Engravings should 
be wetted, and placed with their face to the pre))ared side of the 
paper, and kept in close contact with it. Leaves should have their 
under surface next the glass. For the camera, the most sensitive 
samples of the muriated papers, made with not less than 100 gr. of 
nitrate of silver to the ounce, are selected. The calotype is still 
more certain. The papers intended for the camera require to be 
very carefully prepared. Glass is used instead of paper, after 
being coated with white of egg, or collodion, with which the com- 
pounds of silver are mixed, or over which they are brushed. 



BE ON ZING. 

Bronzing Sculpture^ Wood, doc. 

Bronze of a good quality acquires, by oxidation, a fine green tint, 
called patina antiqaa. Corinthian brass receives, in this way, a 
beautiful clear green color. This appearance is imitated b}^ an 
artificial process, called bronzing. A solution of sal-ammoniac and 
salt of sorrel in vinegar is used for bronzing metals. Any number 
of layers may be applied, and the shade becomes deeper in propor- 
tion to the number applied. For bronzing sculptures of wood, 
plaster figures, (fee, a composition of yelloAv ochre, Prussian blue, 
and lamp-black, dissolved in glue- water, is employed. 



Mix. 



Bronze. 

1. Copper 83 parts; zinc 11 parts; tin 4 parts; lead 2 parts, 
ix. 

2. Copper 14 parts; melt, and add zinc 6 parts; tin 4 parts. 

Ancient Bronze. 
Copper 100 parts; lead and tin each 7 parts. Mix. 



348 Enamels. 

To give an Antique Appearance to Bronze Figures. 
Salt of sorrel 1 part; sal ammoniac 4 parts ; white vinegar 224 
parts. Dissolve, and apply with a camel-hair pencil, just sufficient 
to damp the bronze, previously warmed. Repeat the operation if 
required. 

Keller s Bronze. \ 

Copper 91 parts ; tin 2 parts ; zinc 6 parts ; lead 1 part. Mix. j 

Bronze Poioder. \ 

Bichloride of mercury 1 part ; borax and nitre each 8 parts ; I 

tutty 16 parts ; verdigris 32 parts; oil to make into a paste. Melt. | 

Beautiful Red Bronze Powder. | 

Sulphate of copper 100 parts ; carbonate of soda 60 parts. Apply I 

heat until they unite into a mass, then cool, powder, and add 

copper filings 15 parts. Well mix, and keep them at a white heat 

for twenty minutes, then cool, powder, and wash and dry. 

Bronzing Fluid for Gu7ift, <Joc. 
Kitric acid sp. gr. 1 % nitric ether, alcohol, muriate of iron, each 
1 part. Mix, then add sulphate of copper 2 parts ; dissolved in 
water 10 parts. 



EKAMELS. 

White Enamel. 

Tin 2 parts ; lead 1 part Calcine, then take of the above oxides 
1 part ; crystal 2 parts ; manganese a small portion. Grind well 
together, fuse, and pour the mass into cold water ; dry, grind again 
to powder, and fuse ; repeat the process four or five times, observ- 
ing great care to prevent any contamination from smoke, or iron, 
or copper. 

Another, 
Arsenic 14 parts; potash 25 parts; nitre 12 parts; glass 13 
parts ; flint 5 parts ; litharge 3 parts. 

Blue EnameL 

Fine paste (not metallic) 10 parts ; nitre 3 parts. Oxide of 
cobalt to color. 

Green Enamel. 

Frit 1 pound; oxide of copper ^ ounce; red oxide of iron 12 
grains. 



Marble Staining, 349 



Muxes of Enaynel Colors. 

1. Flint powder 1 part; calcined borax 1 part; flint glass 3 
parts; red lead 4 parts. Keep them in a state of fusion, in a 
Hessian crucible, for three hours; then pour into cold water, dry, 
and powder. 

2. Glass powder 1 1 parts ; white arsenic 1 part ; nitre 1 part. 
Mix. 

Yellow Enamel, 
White oxide of antimony 1 part ; white lead 2 parts ; alum and 
sal-ammoniac each 1 part. Mix in fine powder, and apply just suf- 
ficient heat to decompose the ammoniac. 

Black Enamel. 
Clay 2 parts ; protoxide of iron 1 part. Mix. 



MAEBLE STAINIls'G. 

To Stain Marble. 

It is necessary to heat the marble hot, but not sufficiently so as 
to injure it, the proper heat being that at which the colors nearly 
boil. 

Blue. — Alkaline indigo dye, or turnsole with alkali. 

Red. — Dragon's blood in spirits of wine. 

Yellow. — Gamboge in spirits of wine. 

Gold Color. — Sal-ammoniac, sulphate of zinc, and verdigris, equal 
parts. 

Green. — Sap green, in spirits, with potash. 

Brown. — Tincture of logwood. 

Crimson. — Alkanet root in turpentine. 

The marble may be veined according to taste. To stain marble 
well is a tedious and difficult operation. 

To Stain White Marble. 

Apply with a brush a strong alcohol tincture, made from the root 
alkanet. 

To Clean Marble. 

Chalk (in fine powder) 1 part ; pumice \ part ; common soda 2 
parts. Mix. Wash the spots with this powder, mixed with a little 
water ; then clean the whole of the stone, and wash off with soap 
and water. 

T'o Extract Oil from Stone or Marble. 
Soft soap 1 part ; Fuller's earth 2 parts ; potash 1 part ; boiling 
water to mix. Lay it on the s23ots of grease, and let it remain for a 
few hours. 






80 



350 Compound Colors ix Dyeing. 



COMPOUND COLOKS IN DYEING, 

Are produced by mixing together two simple ones ; or, wliicli is 
the same thing, by dyeing cloth first of the simple color, and then 
by another. These colors var^^ to infinity, according to the propor- 
tions of the ingredients emplo3'ed. From blue, red, and yellow, 
red-olives, and greenish-2:re\'s are made. 

From blue, red, and brown, olives are made from the lightest to 
the darkest shades ; and by giving a greater shade of red, the 
slated and lavender-greys are made. 

From blue, red, and black, greys of all shades are made, such as 
sage, pigeon, slate, and lead-greys. The king's or prince's color is 
duller than usual; this mixture produces a variety- of hues or colors 
almost to infinity. 

From yellow, blue, and brown, are made the goose-dung and 
olives of all kinds. 

From brown, blue, and black, are produced brown-olives, and 
their shades. 

From the red, yellow, and brown, are derived the orange, gold 
color, feuillemort or faded leaf, dead carnations, cinnamon, fawn, 
and tobacco, by using two or three of the colors as required. 

From yellow, red, and black, browns of every shade are made. 

From blue and yellow, greens of all shades. 

From red and blue, purples of all kinds are formed. 

Dyer's Spirit. 
Aquafortis 10 parts; sal-ammoniac 5 parts; tin 2 parts. Dissolve. 
Japan Grounds. 

Red. — Yermillion. makes a fine scarlet, but its appearance in 
japanned work is much improved by glazing it with a thin coat of 
lake, or even rose pink. 

Yellow. — King's yellow, turpeth mineral, and Dutch pink, all form 
very bright yellows, and the latter is very cheap. Seed-lac varnish 
assimilates with yellow very well; and when they are required 
very bright, an improvement may be efi'ected by infusing turmeric 
in the varnish which covers the ground. 

Green. — Distilled verdigris laid on a ground of leaf gold pro- 
duces the brightest of all greens ; other greens may be formed by 
mixing King's yellow and bright Prussian blue, or turpeth mineral 
and Prussian blue, or Dutch pink and verdigris. 
1 Blue. — Prussian blue, or verditer glazed with Prussian blue or 
smalt. 

White. — White grounds are obtained with greater difficulty than 
any other. One of the best is prepared by grinding up flock-white, 
or zinc-white, with one sLxth of its weight of starch, and drying it; 
it is then tempered, like the other colors, using the mastic varnish 
for common uses; and that of the best copal for the finest. Par- 



i^OLISHES. 351 



ticular care should be taken that the copal for this use be made of 
the clearest and whitest pieces. Seed-lac may be used as the 
uppermost coat, where a very delicate white is not required, 
taking care to use such as is least colored. 

Black. — Ivory-black, or lamp-black ; but if the lamp-black be 
used it should be previously calcined in a closed crucible. Black 
grounds may be formed on metal, by drying linseed oil only, when 
mixed with a little lamp-black. The work is then exposed in a 
stove^ to a heat which will render the oil black. The heat should 
be low at first, and increased very gradually, or it will blister. 
This kind of japan requires no polishing. It is extensively used for 
defending iron articles from rust. 



POLISH ESr 

To Polish Brass Inlaid Work. 

File the brass very clean with a smooth file ; then take some 
tripoli powdered very fine, and mix it with the linseed oil. Dip in 
this a rubber of hat, with which polish the work until the desired 
effect is obtained. 

If the work is ebony, or black rosewood, take some elder-coal 
powdered Yary fine, and apply it dry after you have done with the 
tripoli, and it will produce a superior polish. 

The French mode of ornamenting with brass diff'ers widely from 
ours, theirs being chiefly water-gilt {or inolu), excepting the flutes 
of column^, (fee, which are ])olished very high with rotten stone, and 
finished with elder-coal. 

To Brass Plates of Copper. 

The plates previously sufl&ciently heated, expose them to the 
fumes of zinc. 

To Clean Brass. 

1. Finely powdered sal-ammoniac ; v/ater to moisten. 

2. Roche alum 1 part; water IG parts. Mix. The articles to be 
cleaned must be made warm, then rubbed with either of the above 
mixtures, and finished with fine tripoli. This process will give 
them the brilliancy of gold. 

To Brass Vessels of Copper. 
Argol 1 part; amalgam of zinc 1 part; muriatic acid 2 parts; 
water to fill the vessel Boil. 

Method of Cleaning Brass Ornaments. 
Brass ornaments that have not been gilt or lacquered may be 
cleaned, and a very brilliant color given to them, by washing them 



352 



Bookbinders* Recipes. 



with alum boiled in strong ley, in the proportion of an ounce to a 
pint, and afterwards rubbing them with strong tripoli. 

French Polish. 
Alcohol 260 parts; copal varnish 13 parts; sandarach (powdered) 
1 part; mastic (powdered) 1 part; shell-lac (poAvdered) 24 parts. 
Mix, and digest in a moderate heat, in a strong close vessel. 

To French Polish. 

The varnish being prepared (shell-lac), the article to be polished 
being finished off as smooth as possible with glass paper, and your 
rubber being prepared as directed below, proceed to the operations 
as follows: The varnish, in a narrow necked bottle, is to be applied 
to the middle of the flat face of the rubber, bv laying the rubber 
on the mouth of the bottle and shaking up the varnish once, as by 
this means the rubber will imbibe tlie proper quantity to varnish a 
considerable extent of surface. The rubber is then to be inclosed 
in a soft linen cloth, doubled, the rest of tlie cloth being gathered 
up at the back of the rubber, to form a handle. Moisten the face 
of the linen with a little raw linseed oil, applied with the finger to 
the middle of it. Placing your work opposite the light, pass 3'our 
rubber quickly and lightly over its surface until the varnish 
becomes dry, or nearly so ; charge 3'our rubber as before with 
varnish (omitting the oil), and repeat the rubbing, until three coats 
are laid on, when a little oil maybe applied to the rubber, and two 
coats more given to it. Proceeding in this Avay, until the varnish 
has acquired some thickness, wet the inside of the linen cloth, before 
applying the varnish, with alcohol, and rub quickly, lightly, and 
uniformly the whole surface. Lastly, wet the linen cloth Avith a 
little oil and alcohol without varnish, and rub as before till dry. 

To make the Rubber . — Roll up a strip of thick woollen cloth 
which has been torn off, so as to form a soft elastic edge. It should 
form a coil from one to three inches in diameter, according to the 
size of the work. 



BOOKBINDERS' RECIPES. 

Japan Coloring, for Leather Book- Cover s^ (be. 
After the book is covered and dry, color the cover with potash- 
water mixed with a little paste, give it two good coats of Brazil 
wash, and glair it. Put the book between wands, allowing the 
boards to slope a little. Dash on copperas water, then with a 
sponge full of red liquid, press out on the back and on different parts 
large drops, which will run down each board, and make a fine 
ghaded red. When the cover is dry wash it over two or three 
times with Brazil wash, to give it a brighter color. 



Bookbinders' Recipes, 353 



Blue Sprinkle for Bookbinders. 
Strong sulphuric acid 8 ounces ; Spanish indigo, powdered, 2 oz. 
Mix in a bottle that will hold a quart, and place it in a water-batli 
to promote solution. For use, dilute a little to the required color in 
a teacup. 

Blue Marble for Books, (tc. 
Color the edges with King's yellow, and when dry tie the book 
between boards. Throw on blue spots in the gnm trough, wave 
them with the iron pin, and apply the edges thereon. 

Broicn Color for Marbling or Sprinkling Books. 

1. Logwood chips 1 part; annatto 1 part; boil in water 6 parts. 
If too light, add a piece of copperas about the size of a pea. 

2. Umber, any quantity. Grind it on a slab with ox gall and a 
little lampblack. Dilute with ale. 

Gold Sprinkle for Books.^ 
Put into a marble mortar half an ounce of pure honey and one 
book of gold leaf, rub them well together until they are very fine, 
add half a pint of clear water, and mix them well together : when 
the water clears, pour it off, and put in more, till the honey is allj 
extracted, and nothing remains but the gold. Mix one grain of 
corrosive sublimate in a teaspoonful of spirits of wine, and when 
dissolved, put the same, together with a little gum-water, to the 
gold, and bottle it close for use. The edges of the book may be 
sprinkled or colored very dark, with green, blue, or purple, and 
lastly with the gold liquid, in small or large spots, very regular, 
shaking the bottle before using. Burnish the edges when dry, and 
cover them with paper to prevent the dust falling thereon. This 
sprinkle will have a most beautiful appearance on extra work ; 
ladies may use it for ornamenting their fancy work, by putting it 
on with a pen or camel's-hair brush, and when dry burnish it with 
a dog's tooth. 

Marble for Leather Book-covers. 
Wash the cover and glair it, take a sponge charged with water, 
liaving the book between Avands, and drop the water from the 
sponge on the different parts of the cover, sprinkle very fine with 
vinegar black, then with brown, and lastly with vitriol water. 
Observe to sprinkle on the colors immediately after each other, and 
to wash the cover over with a clean sponge and water. 

Chinese Edge for Books. 

1. Color the edge with light licpiid blue and dry; then take a 
sponge charged with vermillion, and dab on spots according to 
fancy ; next throw on rice, and finish the edge with dark liquid ! 
blue. ! 

2. Color light blue on different parts of the edge with a sponge ; 
do the same where there are vacancies with yellow and Brazil red; 

_ .- 



354 



Bookbinders' Recipes. 



I dr}^ and dab on a little Vermillion in spots ; then throw on rice, and 
finish with a bold sprinkle of dark blue. Burnish. 

Wax Marble for Leather Book-covers, (tc. 

This marbling must be done on the fore edge, before the back of 
j the book is rounded, or becomes round, when in boards, and finished 
on the head and foot Take beeswax and dissolve it over the fire 
in an earthen vessel ; take quills stripped of their feathers, and tie 
them together ; dip the quill-tops in the wax, and spot the edge, 
with large and small spots; take a sponge charged Avith blue, 
green, or red, and smear over the edge ; when done, dash off the 
wax, and it will be marbled. This will be useful for stationery 
work, or for folios and quartos. 

Egyptian Marble fjv Leather Book-covers. 

1. Yelloio. — Boil quercitron bark v>'ith water and a little powdered 
alum, over a slow tire, until it is a good strong yellow. Pour the 
liquid into a broad vessel, sufficiently large to contain the cover 
when extended. Before the liquid is cool, take the dry cover, and 
lay the grain side flat on the color; press it lightly that the whole 
may receive the liquid ; let it soak some time, and then take it 
from the vessel. The book must be covered in the usual manner, 
and permitted to dry from the fire. Glair the book ; when dry, 
place it between the wands ; take a sponge and water, and press 
large spots thereon; dip a quill-top into the vinegar black, with it 
touch the water on the cover in different parts, which will have a 
fine effect when managed with care. Let it stand a few minutes, 
then take off the water with a clean sponge. 

2. — Gree-.L — Color the cover in a large A'essel, as mentioned 
before, with Scott's liquid blue ; when done, put it into a vessel of 
clear water for an hour. Take it out and press out the water, then 
cover the book. Glair the cover ; when dry, place it between 
wands, and drop weak potash water from a sponge thereon ; dip 
the quiU-top into the strong black, and touch the water with it 
This must be repeated till you have a good black. "When dry, 
clear it with a sponge and water. 

3. Red. — Boil Brazil dust in rain-water on a slow fire, with a 
little powdered alum and a few drops of solution of tin, till a good 
color is produced. Dip a piece of calf leather into the liquid, and 
you may ascertain the color wanted. If too light, let it boil till it 
is reduced to one half of the quantity ; take it from the fire, add a 
few more drops of the solution of tin, and pour it into a large 
vessel. Put the drv cover on the liquid, and let it remain for a 
quarter of an hour, then press out the water. Color it over with a 
sponge and the quercitron bark water, and cover the book. Glair 
the cover, place it between wands, dash on water with a brush, 
also potash water ; and, lastly, finish it with the strong vinegar 
black, with the quill-top. Observe that too much black is not put 



Bookbinders* Recipes. 355 



on; the intention of the marble is to show the red as transparently 
as possible. 

French Marble for Books. 

Provide a wooden trough, two inches deep, six inches wide, and 
the length of a super-royal sheet. Boil in a brass or copper pan 
any quantity of linseed and water, until a thick mucilage is turnied : 
strain it into the trough and let it cool ; then grind on a marble 
slab any of the following colors in small-beer : Prussian blue, king's 
3'ellow, rose pink, Vermillion, flake white, lamp-black, brown 
umber, green, blue, and yellow, orange, red, and yellow, purple, 
red, and blue, brown, black, and yellow, or red. 

The lamp-black and umber must be burnt over the Are to deprive 
them of their greasy nature. 

For each color you must have two cups, one for the color after 
grinding, the other to mix it with ox-gall, whicli must be used to 
thin the colors at discretion. If too much gall is used the colors 
will spread; when they keep their place on the surface of the 
trough, when moved with a quill, they are fit for use. 

To prevent the water entering between the leaves of the book, tie 
it tight between cutting-boards of the same size, and place the 
trough in a steady situation, to prevent the colors from moving. 

Having all things in perfect readiness for marbling, supposing 
you begin with the blue, throw on with the brush bold spots of 
blue, sprinkle very fine with the white on the blue spots, fill up the 
spaces with red and yellow, by dipping first the quill-top into the 
yellow, and touching the gum therewith, then with the red. The 
red and yellow may be waved or drawn round the blue spots with 
an iron pin, or as the marbler may think proper, according to 
fancy. 

Hold the book with its edge downwards, and press it lightly on 
the colors so disposed on the gum, and the edge will be immediately 
marbled. The colors that may remain on the gum must be taken 
ofi^, .by applying paper thereon, before you prepare for marbling 
again. In this manner you may marble the edges to resemble the 
end-papers, which will have a pleasing eff'ect. 

Chinese Marble for Leather Book-covers, dec. 

Color the cover of the book dark brown, and when dry put it 
into the cutting-press, with the boards perfectly flat; mix whiting 
and water of a tliick consistence and throw it on, in sj^ots or 
streaks, some large and some small, which must remain till dry. 
Spot or sprinkle the cover with liquid blue, and lastly tlirow on 
large spots of licpiid red. The colors must be dry before washing 
off the whiting. 

Orange Sprinkle for Books. 
Color the edge with King's yellow, mixed in weak gum-water, 
then sprinkle witli vermillion mixed in the same manner. 



356 Bookbixdp:rs' Recipes. 



Green Spmikle for Books. 

'1. Yellow tlie edge, then sprinkle with dark blue. 

2. French berries 1 part ; soft water 8 parts. Boil, and add a 
little powdered alum ; then bring it to the required shade of green, \ 
by adding liquid blue. 

Green Marble for Leather Book-covers, cjcc. 

Tlie edge must be marbled with a good bright green only. When 
the color is prepared with the ox-gall, and ready for use, a few 
drops of sweet oil must be mixed therein, the color thrown on with 
a brush, in large spots, till the gum is perfectly covered. The oil 
will make a light edge round each spot, and have a good effect. 

Blue, green, and brown may be also used separately in like 
manner. 

Sheets of paper may be done, having a trough large enough, and 
the sheets damped as for printing, before marbling. 

Spirits of turpentine ma}' be sprinkled on the colors, which will 
make white spots. 

Bmders Thread Marble. 

Yellow the edge ; when dry, cut pieces of thick thread over the 
! edge, which Avill fall on different parts irregularly ; give it a fine 
! dark sprinkle, and shake off the thread. 

nice Marble, for Leather Book-covers, 6:c. 
Color the cover with spirits of wine and turmeric, then place on 
rice in a regular manner ; throw on a very fine sprinkle of cop- 
peras-water till the cover is nearly black, and let it remain till dry. 
The cover may be spotted with the red liquid or potash-water, 
very freely, before the rice is thrown off the boards. 

Orange Color for Marbling or Sprinkling Books, d'c. 
Ground Brazil-wood 16 parts; annotto 4 parts; alum, sugar, 
and gum-arabic, each 1 part ; water 10 parts. Boil, strain, and 
bottle. 

Tree Marble, for Leather Book-covers.. 
A marble in the form of trees may be done by bending the 
boards a little on the centre, using the same method as the common 
marble, having the cover pr.eviously prepared. The end of a 
candle may be rubbed on different parts of the boards, which Avill 
form knots. 

Vinegar Black for Bookbinders, d:c. 
Steep iron filings or rusty iron in good vinegar for two or three 
days, then strain off the liquor. 

To Sprinkle Books, 
Take a stiff brush made of hogs' bristles, perfectly clean, dip it 
in the color; squeeze out the superfluous liquid; then rub a 



Bookbinders' Recipes. S6^ 

_ ^ ^ , ^, 

folding-stick across the brush, and a fine sprinkle will fall on the ' 
edge of the book, which should be previously screwed tight in the 
cutting-press. Repeat the operation until the color is thrown 
equally on every part of the leaves. The brush should be held ini 
the left hand, and the stick in the right. < 

Purple Sprinkle for Bookbinders, 
Logwood chips 4 parts ; powdered alum 1 part ; soft water 24 
parts. Boil until reduced to sixteen parts, and bottle for use. j 

2. Brazil dust (fine), and mix it Avith potash-water for use. | 

Soap Marble for Books. ' 

This is applicable for marbling stationery, book edges, or sheets 
of paper for ladies' fancy Avork. | 

Grind, on a marble slab, Prussian blue, with water, and a little • 
brown soap, to a fine pliable consistence, that it may be thrown on \ 
with a small brush. _ j 

Grind King s yelloAV, in the same manner, with water and Avhite : 
soap. i 

When green is intended for the ground color, grind it with brown 
soap, and King's yellow with Avhite soap. Lake may be used for a 
ground color, and Prussian blue ground with white soap ; brown 
umber for a ground color, and flake-white ground with white soap. : 
Any color of a light substance may be grooind for marbling. | 

Spotted Marble for Books, d:c. | 

After the fore-edge of the book is cut, let it remain in the press, : 
and throw on linseeds in a regular manner ; sprinkle the edge with ; 
any dark color, till the Avhite paper is covered, then shake off the ; 
seeds. Various colors may be used. The edge may be colored i 
with yellow or red before throwing on the seeds and sprinkling 
with blue. The seeds will make a fine fanc}?^ edge when placed | 
very thick on different parts, with a few slightly thrown on the ' 
spaces between. 

Brown Sprinkle for Leather Book-covers, <kc. 

Pearlash or potash 1 part; soft water 4 parts. Dissolve and 
strain. 

Red Sprinkle for Binders, 
Brazil-wood (ground) 4 parts; alum 1 part; vinegar 4 parts; 
water 4 parts. Boil until reduced to seven parts, then add a small 
quantity of loaf-sugar and gum. Bottle for use. 

Black Sprinkle for Leather Book-covers, d'c. 
Green copperas 1 part ; soft water, hot, C parts. Dissolve. 

Stone Marble for Jjcather Book-covers, <$:c. 
Glair the cover, and when dry put the book into the cutting- 
press, with the boards sloping, to cause the colors to run gently 



358 



Crayons. 



down. Throw on weak copperas-water with a brush ; dip a sponge 
into the strong potash-water, and press out the color from the 
sponge on different parts of the back, so that the colors niay run 
down each side fr-om the back. Where the brown has left a 
vacancy apply vitriol-water in the same manner. The book must 
remain till perfectly dry before washing it. 



c E A T o N s . 

Litliographic Crayons. 

1. Take wliite wax 4 parts; gum-lac 2 parts. Melt over a gentle 
fire, then add dry tallow soap in shavings 2 parts. Stir until dis- 
solved. K'ext add white tallow 2 parts; copal varnish 1 part; 
lampblack 1 part. Mix well, and continue the heat and stirring 
until, on trial by cooling a little, it appears of a proper quality, 
which should be that it will bear cutting to a fine point, and trace 
delicate lines without breaking. 

2. Take dry Avhite tallow soap 6 parts ; white wax 6 parts ; 
lampblack 1 part. Fuse in a covered vessel. 

3. Take lampblack 1 part; tallow soap 2 parts; shell-lac 2 
parts ; wax 4 parts. Mix, with heat, and mould. 

4. Take dried tallow soap 5 parts ; wax 4 parts ; lampblack 1 
part. Mix as before. 

Crayons. 

1. Shell-lac 6 parts; spirit 4 parts; turpentine 2 parts; color 12 
parts; pale clay 12 parts. Mix. 

2. Pipe-clay, color as required, water to mix. Form into a stiff 
paste, and roll it into crayons. 

To Fix Crayon Colors, 
Paste j-our paper on canvass, in a frame, in the usual v/ay, then 
brush over the back two or three times with the following mixture, 
and w4ien the last coat is dry give the face of the picture one or 
two coats in the same way. This will make it resemble an oil 
painting. Spirits of turpentine 10 parts ; boiled oil 6 parts. Mix. 

To render permanent Chalk or Pencil Drawings. 
La}' the drawing on its face, and give the back two or three thin 
coats of the following (No. 1) mixture ; let it dry, and turn it with 
the chalk upwards, and give that side one or two coats also ; lastly, 
if you choose, give it one or two coats of No. 2. 

1. Isinglass or gum-arabic 5 parts ; water 12 parts. Mix. 

2. Canada balsam 4 parts ; turpentine 5 parts. Mix. 

Wash to fix BlacTclead Pencil Drawings, 
1, Isinglass 1 part; water 50 parts. Dissolve with heat^ and 

filter. 



Gilding, 859 

2. Take skimmed milk, and strain. For use, pour the liquid on a 
surface sufficiently large, and take the drawing by the corners, lay 
it flat on the wash, then carefully remove it, and place it on a 
slanting surface to drain and dry. "This will also answer for chalk 
drawings. 



GILDING. 

To Gild or Silver Leather. 
Finely powder resin, and dust it over the surface of the leather, 
then lay on the leaf, and apply (hot) the letters or impression you 
wish to transfer ; lastly, dust off the loose m^etal with a cloth. The 
cloths used for this purpose become^ in time, very valuable, and are 
often sold to the refiners for $5 to $1. 

To gild on Calf and Sheep Skin. 

Wet the leather with the white of eggs ; when dry rub it with 
your hand and a little olive oil, then put the gold leaf, and apply 
the hot iron to it. Whatever the hot iron shall not have touched 
will go off by brushing. 

To gild Copper, Brass, &c. {Patent.) 
Fine gold 5 parts; nitric acid (sp. g. 1*45) 21 parts; h3'dro- 
chloric aci(} (sp. g. 1*1 5) 17 parts; pure water 14 parts. Digest 
with heat in a glass vessel until all the gold is dissolved, and till 
red or yellow fumes cease to rise. Decant the clear liquid into 
some convenient vessel, and add water, 500 to 600 parts. Boil for 
two hours, let it stand to settle, and pour off the clear into a 
suitable vessel. For use, heat the liquid and suspend the articles 
(previously well cleaned) by means of a hair or fine Avire, until suf- 
ficiently coated with gold, then well wash them in pure water. 

To gild Glass and Porcelain. 

1. Apply to the part a surface of gold size ; when nearl}^ diy lay 
on the leaf. 

2. Gold powder 2 parts; borax 1 part; turpentine to mix. Mix 
and apply to the surface to be gilded with a camel-hair pencil; 
when quite dry, heat it in a stove until the borax vitrifies. Bur- 
nish. Platina, silver, tin, bronze, cfcc, may be applied in a similar 
manner. 

To give Iron the color of Copper. 
Take 1 oz. of copper-plates, cleansed in the fire ; 3 oz. of aqua- 
fortis; dissolve the copper, and when it is cold use it by wasliins; 
your iron with it by tlic help of a feather; it is presently' cleansed 



360 GiLDIXG. 

I 

: and smooth, and will be of a copper color; by much, using or 

I rubbing it will wear off, but may be renewed by the same process. 



A ii:ay of Gilding icith Gold upon Silver. 
Beat a ducat thin, and dissolve in it two ounces of aqua-regia ; 
dip clean rags in it, and let them dry ; burn the rags, and, with the 
tinder thereof, rub the silver with a little spittle ; be sure first that 
the silver be cleansed from grease. 

Gilder s IVax. 

1. Yellow wax 3 pounds; verdigris 1 pound; sulphate of zinc 1 
pound ; red oxide of iron 2^ pounds. Powder the last three 
articles very fine. 

2. Yellow wax Y pounds ; colcothar 7 pounds ; verdigris 3 
pounds ; borax J pound ; alum J pound. 

To dye in Gold Silver Medals, or Laminas, through and through. 

Take glauber salt, dissolve it in warm water, so as to form a 
saturated solution. In this solution put a small proportionate 
quantity of calx, or magister of gold. Then put and digest in it 
silver laminas cut small and thin, and let them lie twenty-four 
hours over a gentle fire. At the end of this term you will find 
them thoroughly dyed gold color inside and out. 

To gild Silks, Satins, d:c. 

Kitromuriate of gold, in solution, 1 part ; distilled water 3 parts. 
Mix. Lay out any design with this fluid, and expose it, while wet, 
to a stream of hydrogen gas; then wash it with clear water. 

To make Transi^arent Silver. 
Kefined silver one ounce ; dissolve it in two ounces of aqua- 
fortis; precipitate it with a pugil (a quantity that may be taken 
up between the thumb and finger) of salt, then strain it tlirough a 
paper, and the remainder melt in a crucible for about half an hour, 
and pour it out, and it will be transparent. 

To make Copper into a Metal like Gold. 
Distilled verdigris 4 oz. ; Tutise Alexandrinse prseparatse two oz. ; 
saltpetre 1 oz. ; borax -^ oz. Mix all together with oil, till they be 
as thick as pap ; then melt it in a crucible, and pour it into a fire- 
shovel, first well warmed. 

3fercurial Plating. 

Quicksilver 4 parts ; nitric acid 4 parts ; finely powdered cream 
of tartar 2 parts ; finely powdered salt of sorrel 1 part. Dissolve 
the silver in the acid, then add the rest, and stir until dissolved. 
This Imparts a pleasing silvery appearance to articles formed of 
eopper, b}^ merely applying it with the finger. 

Grecian Gilding. 

Take sal-amaaiojiiac and bichloride of mercury, equal parts, 



Glass Stains. $Ii1 



dissolve in nitric acid, and make a solution of gold with this fluid, 
lay it on the silver, and expose it to a red heat ; it will then be 
gilded. 

To gill or silver Writing. 
Let there be a little gum and lump-sugar in the ink you write 
with ; when dry, breathe on it and apply the leaf. 

To whiten Copper throughout 
Take thin plates of copper, as thin as a knife, heat them six or 
seven times, and quench them in water ; then melt them, and to 
each pound add 4 ounces of saltpetre and 4 ounces of arsenic, well 
powdered and mixed, and first melted apart in another crucible, ; 
by gentle degrees; tlien take them out, and powder them; then! 
take Venetian borax and white tartar, of each an ounce and a half;! 
then melt these, with the former powder, in a crucible, and pour themj 
out into some iron receiver ; it will appear as clear as cr^^stal, and ! 
is called cryatallinuin fixum arsenicum. Of this clear matter, broken! 
into little pieces, throw into the melted copper (by small pieces atj 
a time, staying five or six minutes between each injection) 4| 
ounces; when all is throAvn in, increase the fire, till all be wellj 
melted together for a quarter of an hour; then pour it out into an- 
ingot. 

To gild Steel. 
Apply an etherial solution of gold. This is equally adapted to! 
lettering, as wholly covering the object. It may be applied with a 
pen, or otherwise. 



GLASS STAINS. 

Red Stain for Glass. 

1. Rust of iron 100 parts; glass of antimony 99 parts; yellow! 
glass of lead 98 parts ; sulphuret of silver 3 parts. Mix. 

2. White hard enamel 100 parts; red chalk 50 parts ; peroxide 
of copper 5 parts. Reduce to fine powder, and mix. 

Blue Glass. 
Plain paste 300 parts ; zaffre 3 parts ; manganese 1 part. If the 
glass should be of too deep a blue, use less zaflVe and manganese ; 
if too purple, omit the manganese altogether. 

Black Slain fur Glass. 

1. Black scales of iron 29 parts ; white cr3-6tal glass 4 parts ; 
antimony 2 parts; manganese 1 part; vinegar to mix. 

2. Glass of antimony 1 part ; oxide of copper 2 parts ; crystal 
glass 3 parts. Mix. 

ST ~" 



362 Factitious Stones. 



Orange Stain for Glass. 
Precipitated silver powder, yellow oclire, red ochre, equal parts. 
Turpentine to mix. 

Brown Stain for Glass. 
White glass 2 parts ; manganese 1 part. Mij^, 

Flesh Color for staining Glass. 

Red lead 1 part ; red enamel 2 parts. Mix with alcohoL 
Yellow Stain for Gla^s. 

Chloride of silver 1 part ; burnt pipeclay 3 parts. Keduce to fine 
powder, and mix. This stain must be applied t©' the- back of the 
glass. 

To Marble a Glass Globe, 

Grind w^ell on a stone, minium for red, tnrmeris, or rather 
cerussa citrina, for yellow, smalt for blue, verdigris for green, 
ceruse, or chalk, for white. Work each in oil separate, and with a 
hog's hair pencil, single or mixed, as j'ou think fit, scatter the same 
into the glass, and roll it, or dispose the colors, as yo-u like. Thecv 
last of all,, fling a little mead amongst them, which ©over& alL 

For the Magic Lantern, paint the glasses with transparent colore 
tempered with oil of spike. 



FACTITIOUS STONES. 

Factitious Amethyst. 

1. Take strass 5000 parts; oxide of manganese 37 parts; oxide 
of cobalt 25 parts ; purple of Cassius 1 part. Fuse for twenty-six 
hours, and cool slowly. 

2. Take paste or strass 10,000 parts; oxide of manganese 25 
parts ; oxide of cobalt 1 part. 

Factitious Emerald, 

i. Oxide of chrome 1 part; green oxide of copper 20 parts; 
strass 2300 parts. Fuse with care for twenty-six hours, then cool 
slowly. 

2. Strass 10,000 parts; acetate of copper 150 parts; protoxide of 
iron 3 parts. As before. 

3. Strass 6600 parts; carbonate of copper 60 parts; glass of anti- 
mony 6 parts. Fuse with care. 

4. Strass 500 parts ; glass of antimony 20 parts ; oxide of cobalt 
3 parts. As before. 

Artificial Coral. 
Yellow resin 4 parts; vermillion 1 part. Melt. This gives a 
very pretty effect to glass, twigs, cinders, stones, &c., dipped into 



Factitious Stones. 363 



it. It is also useful for a cement for ladies' fancy work, such as 
grottoes, cfec. 

Paste resemhling the Red Coryielian. 
Plain paste 1000 parts; glass of antimony 500 parts; calcined 
vitriol 63 parts or less ; manganese 4 parts. Melt together. 

Pade resembling the White Cornelian. 
Plain paste 1000 parts; yellow ochre 8 parts; calcined bones 31 
parts. As before. 

Factitious Opal. 

1. Strass 500 parts; horn silver 10 parts; calcined magnetic ore I 
2 parts ; chalk marl 25 parts. Mix in fine powder, and fuse with \ 
great care. i 

2. Plain paste 100 parts ; calcined bones 6 parts. j 

Factitious Oriental Ruhy. i 

Strass 7000 parts; precipitate of Cassius and nitric peroxide ofj 
iron each 165 parts; golden sulphuret of antimony 160 parts;! 
manganese calcined with nitre 150 parts; rock crystal 1000 parts. 
Mix in fine powder, and carefully melt. 

Factitious Sapphire. 

1. Oxide of cobalt 1 part ; strass 80 parts. j 

2. Paste or strass 2300 pai-ts ; oxide of cobalt 34 parts. Fuse j 
carefully for thirty hours. i 

3. Plain paste 100 parts; smalts 12 parts; manganese 1 part. As! 
before. i 

4. Plain paste 10 pounds; zaffre 3 drachms; precipitate of gold- 
and tin 1 drachm. As before. | 

Factitious Topaz. \ 

1. Strass 1000 parts; glass of antimony 42 parts; purple of: 
Cassius 1 part Fuse for twenty-four hours, and cool slowly. 
% Strass 4000 parts ; saff^ron of Mars 40 parts. As before. 

To solder together Rubies. 
Apply them to a strong flame by means of the blow-pipe, and 
when sufficiently soft unite them with care ; they will neither lose 
their color nor weight. 

Factitious Ruby. 

Strass 40 parts; oxide of manganese 1 part. Mix, and treat as 
for topaz. 

White Crystal, or Factitious Diajnond. 

Manganese 1 part; rock crystal 2800 parts; borax 1900 parts; 
white lead 5700 parts. Mix in fine powder, then fuse in a clean 
crucible, pour it into water, dry, })owder, and repeat the process 
two or three times. 



364 FACTiTiors Stone?. 



Compos'dio7i for Fixed Brilliants. 
Meal gunpowder IG parts; zinc, or steel, or cast-iron horinsrs G 
parte. Mix, 

PafUe resembling Vinegar Garnet. 
Plain paste 1000 parts ; glass of antimony 500 parts ; calcined 
iron 16 parts. Add the antimony last:. 

Gold or Yellow Paste. 

Take plain paste (made without the saltpetre) 100 parts; oxide 
of iron 1 part. Fuse. 

Factitious Lapiz Lazuli. 

Plain paste 1000 parts ; calcined bones V3 parts; zaffre T parts; 
magnesia 5 parts. If it is desired to vein it witli gold — gold pow- 
der and borax, equal parts ; vein the cakes to taste, and then heat 
them sufficiently hot for cementation. 

Foils for Cri/stafs, Pastes, 

Put two or three layers of tin-foil into the socket made for the 
stone, heat it gently, and fill it with quicksilver, let it rest two or 
three minutes, then pour it out, and place in the stone. 

Factitious Yellow Diamond. 
Strass 50<j parts ; glass of antimony 10 parts. Juse. 

Another. 
Strass 500 parts ; chloride of silver 25 parts. Mix, and fuse. 
• "- Strass, or Jfaycnce Base. 

1. Pure rock crystal or flint, S parts ; salt of tartar 25 parts. 
Powder, mix well, bake, and cool, then put it into a basin of 
water, and add dilute nitric acid until effervescence ceases ; collect, 
wash, and dry the powder; next add fine white-lead 12 parts. 
Levigate and well wash it with pure water, then of the above mix- 
ture dried 12 parts; calcined borax 1 part. Triturate them toge- 
ther, melt in a clean crucible, and pour the mixture into cold water; 
dry, powder, and melt it in the same manner, a third time, alwa^'s 
in' a fresh crucible, observing to separate any lead that may be 
revived. To the third frit, ground to powder, add purified nitre 
|- part. Remelt, and a mass of crystal will be found in the crucible 
of a beautiful and diamond-like lustre. 

2. Ai-senic 1 part; borax 23 parts; pure pearlash 180 parts; 
minium 525 parts ; rock crystal 338 parts. Mix, as before. 

3. Arsenic 1 part; borax 30 parts ; pota.*h 105 parts; carbonate 
of lead 709 parts; fine white sand 815 parts. Mix with care. 

4. Arsenic 1 part; borax 35 parts; potass 325 parts; minium^ 
900 parts ; rock crystal 580 parts. Treat as before. 

5. Rock crystal 400 parts ; pure white lead 945 parts ; pare 
potash 140 parts ; borax 41 parts. 



Inks. 365 

6. Pure potash 2 parts; fine white sand 15 parts; litharge 20 
parts. See also Paste. 



INKS. 

Indestructible Ink, 

1. Powdered copal 25 parts ; oil of lavender 200 parts ; lamp- 
black 2 parts; indigo 1 part. Dissolve. 

2. Asphaltum 1 part; lamp-black i part. Melt, then add oil 
prepared for printers' ink, by boiling and burning until sufficiently 
stringy, 1^ part. Mix together, and add spirits of turpentine 3 or 
4 parts. We would propose this ink, made with less turpentine, so 
as to be sufficiently thick for stamping, as the most perfect pre- 
ventive of fraud, as when applied to the surface of an engraving, or 
letter-press, nothing will remove it that will not also discharge the 
ink of the stamp. It will stand the action of the alkalies, chlorine, 
acids, <&c., even in a heated state, when they will at once destroy 
the texture of the paper. 

Lithographic Inh. 

1. Take Venice turpentine 1 part; lamp-black 2 parts; tallow 6 
parts ; hard tallow soap 6 parts ; mastic in tears 8 parts ; shell- 
lac 12 parts; wax 16 parts. Melt, and pour it out on a slab. 

2. Take dry tallow soap 5 parts ; mastic in tears 5 parts ; Scotch 
soda 5 parts; shell-lac 25 parts; lamp-black 2 parts. Fuse the 
soap and lac, then add the remainder. 

For use, this ink must be rubbed down with water in a saucer 
(warmed), until an emulsion is formed of a proper consistence to 
flow easily from a pen or pencil. 

Blue Writing Fluid. 

1. Ferrocyanide of iron, powdered, and strong hydrochloric 
acid, each 2 parts. Dissolve, and dilute with soft water. 

2. Indestructible. — Shell-lac 4 parts ; borax 2 parts ; soft water 36 
parts; boil in a close vessel till dissolved; then filter, and take of 
gum-arabic 2 parts; soft water 4 parts. Dissolve, and mix the twoj 
solutions together, and boil for five minutes as before, occasionally; 
stirring to promote their union; when cold, add a sufficient quan-j 
tity of finely powdered indigo and lamp-black to color ; lastly, let 
it stand for two or three hours, until the coarser powder has sub- 
sided, and bottle for use. Use this fluid with a clean pen, and keep 
it in glass or earthen inkstands, as many substances will decom- : 
pose it while in the liquid state. When dry, it will resist the' 
action of water, oil, turpentine, alcohol, diluted sulphuric acid, ! 
diluted hydrochloric acid, oxalic acid, chlorine, and the caustic 
alkalies and alkaline earths. 



31* 



866 Inks. 

Red Ink for Writing. 
Boil over a slow fire 4 ounces of Brazil wood, in small raspings 
or chips, in a quart of water, till a third part of the water is eva- 
porated. Add during the boiling 2 drachms of alum in powder. 
When the ink is cold steam it through a fine cloth. Vinegar or 
stale urine is often used instead of water. In case of using water 
adding a very small quantity of sal-ammoniac would improve this 
ink. 

Fine Black Writing Ink. 
Take 2 gallons of a strong decoction of logwood, well strained, 
and then add 1^ pounds blue galls in coarse powder; 6 ounces 
sulphate of iron ; 1 ounce acetate of copper ; 6 ounces of well 
ground sugar; and 12 ounces of gum-arabic. Set the above on the 
fire until it begins to boil, then set it away until it has acquired the 
desired black. 

Black Ink imprcyved. 
To I pint of common black ink add 1 drachm of impure car- 
bonate of potassa, and in a few minutes it will be a jet black. Be 
careful that the ink does not run over, during the effervescence 
caused by the potassa. 

Green Ink. 

1. Cream of tartar 1 part; verdigris 2 parts; water 8 parts. 
Boil until reduced to a proper color. 

2. Crystallized acetate of copper 1 ounce ; soft water 1 pint. Mix. 

Marking Ink. 

Lunar caustic 2 parts ; sap green and gum-arabic each 1 part ; 
distilled water. Dissolve. 

The Preparation. — Soda 1 ounce ; water 1 pint ; sap green \ 
drachm. Dissolve, and wet the linen (where you intend to write) 
with this mordant, then well dry it. 

Printing Ink. 

1. (Very fine.) — Balsam of capaivi 9 parts; fine lamp-black 4 
parts; indigo 1 part; dry yellow soap 3 parts. Grind perfectly 
smooth. 

2. (Extemporaneous.) — Balsam of capaivi, lamp-black to color. 
Grind well together with a little soap. 

3. Take linseed oil ; heat it in a proper vessel until it begins to 
boil, then remove it from the fire, and kindle the vapor ; allow it 
to burn till it becomes stringy when tried between the fingers, then 
add gradually to every quart black resin 1 pound. Dissolve, and 
add yQYj cautiously dry brown soap in shavings, 4|^ ounces to every 
quart. Set it upon the fire, and stir the mixture until the combi- 
nation is complete; next, put into a suitable pot, finely ground 
indigo 1 ounce; fine Prussian blue 1 ounce; fine lamp-black 18 



Inks. S6l 

ounces. For every pound of resin employed pour the liquid on the 
color, well mix, and lastly, subject it to the action of a mill. 

Indelible Ink for Marking Linen. 

1. The juice of sloes 1 pint ; gum -^ ounce. This requires no mor- 
dant, and is very durable. 

2. Nitrate of silver 1 part; water 6 parts; gum 1 part. Dissolve. 
If too thick dilute with warm soft water. 

Autographic Ink for Lithographers. 
White soap 25 parts ; white wax 25 parts ; mutton suet 6 parts ; 
lamp-black 6 parts; shell-lac 10 parts; mastic 10 parts. Mix with 
heat, and proceed as for lithographic ink. 

To restore Writing effaced with Chloride. 

1. Expose it to the vapor of sulphuret of ammonia, or dip it into 
a solution of the sulphuret. 

2. Ferrocyanide of potass 5 parts; water 85 parts. Dissolve, 
and immerse the paper in the fluid, then slightly acidulate the solu- 
tion with sulphuric acid. 

To give an appearance of Age Co Writijig. 
Infuse a drachm of saffron in half a pint of ink, then write with 
it. 

Perpetual Ink for Tombstones^ Marble^ <kc. 

Pitch 11 parts; lamp-black 1 part; turpentine sufficient Mix, 
with heat. 

Blue Ink. 
Take sulphate of indigo, dilute it with water till it produces the 
color required. It is with sulphate very largely diluted, that the 
faint blue lines of ledgers and other account books are ruled. If 
the ink were used strong, it would be necessary to add chalk to it 
to neutralize the acid. The sulphate of indigo may be had of the 
woollen dyers. 

Copying Ink. 
Add 1 ounce of moist sugar to every pint of common ink. 

Red Permanent Ink. 
Vermillion 4 parts ; sulphate of iron 1 part ; drying oil to mix. 
Any other color will answer besides red. This ink will resist most 
of the usual reagents. 

Black Permanent Ink. 

Nitrate of silver 2 parts ; distilled water 28 parts ; sap green 1 
part. Dissolve. 

For the Mordant. — Common soda 2 parts; gum-arabic 1 part; 
soft water 8 parts. Mix, and moisten the linen with this fluid, and 
well dry before using the ink. 



368 Inks. 



Yellow Ink. 



1. French berries 1 pound ; alum 2 ounces ; water 1 gallon. 
Boil and strain, then add gum-arabic 4 ounces. 

2. Water 80 parts; Avignon berries 7 parts ; gum and alum each 
5 parts. Boil for one hour, and strain. 

Blue hik for Ruliyig. 

Take 4 ounces of vitriol, best quality, to 1 ounce of indigo; pul- 
verize the indigo very fine ; put the indigo on the vitriol, let themj 
stand exposed to the air for six days, or until dissolved; then fillj 
the pot with chalk, add half a gill of fresh gall, boiling it before 
use. 

Black Ink for Ruling. 

Take good black ink, and add gall as for blue; do not cork it, as 
it will prevent it from turning black. 

Red hik for Ruling. 

One pound of Brazil wood to one gallon of the best vinegar ; let 
tlie vinegar simmer before 3'ou add the wood, then let them simmer 
together for half an hour, then add three quarters of a pound of 
alum to set the color ; strain it through a woollen or cotton cloth, 
cork it tight in a stone or glass bottle. For ruling, add half a gill 
of fresh gall to 1 quart of red ink, then cork it up in a bottle for 
use. 

Indian Ink. 

1. Take finest lamp-black, and make it into a thick paste with 
thin isinglass; size, then mould it; attach the gold leaf, and scent 
with a little essence of musk. . 

2. Take lamp-black, make it mto a thick paste with gum-water, 
and mould it. 

Carbon Ink. 

Dissolve real India ink in common black ink; or add a small 
quantity of lamp-black, previously heated to redness, and ground 
perfectly smooth with a small portion of the ink. 

Gold and Silver Ink. 
Fine bronze powder, or gold or silver leaf, ground with a little 
sulphate of potash, and washed from the salt, is mixed with water 
and a sufficient quantity of gum. 

Gluten Ink. 
Dissolve wheat gluten, free from starch, in weak acetic acid of 
the strength of common vinegar; mix 10 gr. of lamp-black and 2 
gr. of indigo with 4 oz. of the solution, and a drop or two of oil of 
cloves. 



Waxes. 369 

Inh for writing on Zinc Labels — Horticultural Ink. ' 

1. Dissolve 100 gr. of chloride of platina in a pint of water. A 
little mucilage and lamp-black may be added. 

2. Sal-ammoniac 1 dr.. verdigris 1 dr., lamp-black | dr., water 
10 dr. Mix. 

Chrome Ink. 
Extract of logwood | oz. ; gum i oz. ; water a pint. Dissolve 
also in 12 oz. of water ^ oz. of yellow chromate of potash (or J oz. 
each of bichromate and bicarbonate of potash). Mix the two solu- 
tions. The ink is ready for immediate use. 

Ink for writing on Steel, Tin Plate, or Sheet Zinc. 
Mix 1 ounce of powdered sulphate of copper and ^ ounce of pow- 
dered sal-ammoniac, with 2 ounces of diluted acetic acid ; adding 
lamp-black or vermillion. 



WAXES. 

Black Sealing-wax. 

1. Shell-lac 2 parts ; yellow resin 3 parts ; ivory black 2 parts. 
Powder fine, and mix by melting carefully. 

2. Yellow resin 15 pounds; lard 1 pound; beeswax 1 pound; 
lamp-black 3 pounds. Mix with heat. 

Soft Sealing-wax. 
Yellow resin 1 part ; beeswax 4 parts ; lard 1 part ; Yenice tur- 
pentine 1 part ; color to fancy. Mix Avitli a gentle heat. 

Gold Colored Sealing-wax. 

1. Bleached shell-lac 1 pound ; Yenice turpentine 4 ounces. 
Melt, and add gold colored talc as required. 

2. Bleached shell-lac 3 pounds; turpentine 1 pound; Dutch leaf, 
ground fine, 1 pound or less. Mix with a gentle heat. The leaf 
should be ground or powdered sufficiently fine without being 
reduced to dust. 

Green Sealing-wax. 
Shell-lac 2 pai'ts ; yellow resin 1 part ; verdigris 1 part. Powder 
and mix by heating slowly. 

Scented Sealing-wax. 

1. Balsam of Peru 2 parts; sealing-wax composition 130 parts. 
Mix, with a gentle lieat 

2. Sealing-wax composition 99 parts ; essence of musk 3 parts. 
Add the latter when the wax is cooling, and stir well. 



370 Waxes. 

3. Wax composition 96 parts; oil of lavender 4 parts; oil of 
lemon 3 parts. As before. 

j Blue Sealing-wax. 

Shell-lac 2 parts ; smalts 1 part ; yellow resin 2 parts. Powder, 
and mix caret nlly with heat. 

Hed Sealing-wax. 

1. Shell-lac 2 parts ; resin 1 part ; vermillion 1 part. Powder 
fine, and melt over a slow fire, 

2. Yellow resin 14 parts ; Venetian turpentine 4 parts ; beeswax 
1 part ; red or orange lead tO parts. Mix, with heat. 

3. Oil of turpentine 1 part ; lard 1 part ; vermillion 2 parts ; 
gum-lac 12 parts. Mix, with a gentle heat 

4. (Very fine.) — Shell-lac 4 parts; Venice turpentine 1 part; 
vermillion 3 parts, Mix. 

Engravers^ Border Wax. 
Beeswax 1 part; pitch 2 parts; tallow 1. Mix. 

Black Bottle Wax. 
Common resin 20 pounds ; tallow 5 pounds ; lamp-black 4 j 
pounds. Mix, with heat. 

Red Bottle Wax. j 

Common resin 15 pounds; tallow 4 pounds ; red lead 5 pounds. | 
Mix, with heat. Any color may be employed. 

Marbled Sealing-wax. 
Take wax of different colors and melt them in separate vessels, 
and when they begin to cool a little atir them all together, and 
form the mass into sticks, 



THE 

ENGINEERS S FIELD BOOK: 

CONTAINING yOEMTJL.E 

FOE THE VARIOUS METHODS OF ELHSTNING AND CHANGING 
LINES, LOCATING SIDE TRACES AND SWITCHES, Etc. 

ANI> 

TABLES 

01^ RADII AND THEIR LOGARITHMS, NATURAL AND LOGAEITHMIG 
VERSED SINES, AND EXTERNAL SECANTS, &c. 

TOGETHER WltH A TABLE OF 

ifATUEAL SlNfiS Atit) TANGENTS, Etc., 

TO £VBRY DEGiSEE AND MINUTE OF THE QUADRANT. 

ANl) LOGARITHMS OF NUMBERS FROM 1 TO 10,000. 

CHARLES HASLETT, 



EECOMMENDATIONS. 



Office of the 0. c£' M. R R. Co., Cincinnati^ May, 1854. 

Having examined Mr. Haslett's Field Book for Eailroad Engi- 
neers, and made use of the rules he has laid down in many instances 
in field work, on the division of which I have had charge, I am 
satisfied of its superiority to an}' similar w^ork yet published, in com- 
prehensiveness and the ready application of the rules. The intro- 
duction of versed sines and external secants into the calculations 
very much reduces the time and labor required by the usual 
method of calculations for locating lines. 

I recommend it to engineers, as being a book combining accuracy 
and a ready application to field practice. 

J. B. CUMMINGS, 

Engineer EmUrn Div. Ohio and Mississippi R. R. 

I most fully concur in recommending Mr. Haslett^s work to the 
attention of Engineers, believing it better than anything of the kind 
yet published. 

K A. GURNEY, 

Chief Engineer, Indiana South Western R. R, 



C. A. Haslett, Rsq. — Dear Sir: — T Lave examined with consider- 
able care the work you propose to publish for the use of engineers 
in the field, and I have no hesitancy in saying that it will be the 
most useful of any work of its character yet offered to the public. 

Yours very truly, 

A, &^ Osgood, 
Divisio7i Engineer, Ohio and Mississippi R. R. 



I concur with Mr. Cummings in the opinion that Mr. Haslett's 
mode of locating lines ver\' much reduces the time and labor 
required by the usual method. 

S. S. Post. 
Chief Engineer, Ohio and Mississippi R. R. 



From statements received from engineers of the Ohio and Missis- 
sippi Railroad who have used Mr. Haslett's method, I have every 
reason to believe it to be an improvement in simplicity and accuracy 
over the old methods commonly in use. 

O. M. Mitchell, 
Con. Engineer, Ohio arid Mississippi R. R, 



From the foregoing recommendation, w^ith a hast}' examination of 
the tables, I concur in the opinion of Messrs. Post <fe Mitchell. 

E. Gest, Engineer, 



Preface. ZlZ 



PREFACE. 



In presenting this work to the public, the Author claims for it the 
adaptation of a new principle in trigonometrical analysis of the 
formulas generally used in field calculations. Experience has 
shown, that versed sines and external secants as frequently enter 
into calculations on curves as sines and tangents ; and by their 
use, as illustrated in the examples given in this work, it is believed 
that many of the rules in general use are much simplified, and 
many calculations concerning curves and running lines made less 
intricate, and results obtained with more accuracy and far less 
trouble, than by any methods laid down in works of this kind. 

The examples given have all been suggested by actual practice, 
and will explain themselves. It has not been thought necessary to 
enter into all the details of demonstration, as this book is int(nled 
expressly for use in the field ; and engineers seldom have time to 
enter into tedious geometrical demonstrations, when direct appli- 
cation of rules is required. 

As a book for practical use in field work, it is confidently believed 
that this is more direct in the application of rules and facility of 
calculation than any work now in use. 

In addition to the tables generally found in books of this kind, 
the author has prepared, with great labor, a Table of Natural and 
Logarithmic Versed Sines and External Secants, calculated to 
degrees, for every minute ; also, a Table of Radii and their Loga- 
rithms, from 1° to 60°. Rules and examples are also given for 
running curves without the use of an instrument; also for locating 
turnouts, side tracks, switches, <fec. 

Having been for several years engaged in eiirveys and locations 
of railroads, and practically convinced of the great saving of time 



S74 Preface. 

and trouble gained by using the rules and principles given in this 
book, the Author submits it, without further preface, to the pro- 
fession, fully confident that its use will be practical proof of its 
merits. 

The tables and examples have been prepared with great care, 
and their accuracy may be relied upon. 

While the Author claims a fair share of originality in the follow- 
ing work, he would acknowledge many valuable suggestions 
derived from Mifflin's Piagrams, as also from Henck on Compound 
and Reversed Curves, authors to whom he would refer those 
wishing to follow the subject at greater length. On the manner of 
working an instrument Mifflin is very clear and concise. This work 
is designed especially for practical field engineers^ already familiar 
with minor details. 

a H, 

Gincinyiatiy l^^^* 



^xJimukx 



T n E 



ENGINEEE'S FIELD BOOK. 



FORMULAE FOR RUNNING LINES, LOCATING SIDE-TRACKS, 

&c. 

PROPOSITION I. Fig. 1.* 

To change the origin of a curve so that it shall terminate in a tangent 
parallel to a given tangent, 

F 




Fy.,. 



Suppose the curve A C to have been described containing 60° of 
curvatiire, and that the distance G D equal 50 feet. 

We have by logarithms : 

Sine 60° (total amount of curvature), . O'OS'TSSl 

Is to R 10-000000 

So is G D, 50 feet 1-698970 

To AB rz 57'78 feet, .... 1-76T439 

r. I . GD 50 ^^„ 

Or by nat. sines = -. = = 67 '7 3. 

^ sin. 60° -86608 

Produce the tangent from A to B =: 57-73 feet; then make the 

* The diagrams in this work are not drawn to any exact scale, but are designed 
to represent merely the abstract geometrical relation of linos. 



376 



Formula for Running Lines, 



j curve B D equal A C ; that is A M C = B X D ; then the tangents ; 
! will be parallel. j 

This rule will apply to the origin of a compound curve, using the : 
total amount of curvature run. 



PROPOSITION II. Fig. 2. | 

Having a curve A B terminating in a tangent D F, it is required to j 
find the radius of a curve that will give a tangent C G parallel to\ 
D F at any given distance therefrom, as at D C say 30 feet, i 




Let AM be the given radius = 1146 feet, the arc AB = 800 
feet, containing 40°, and D C perpendicular distance 30 feet. 
By logarithms : 

As versed sine 40° . . . . 9-369133 

Is to R 10-000000 

So is D C =r 30 feet .... 1-477121 

To M iS'= difference of radii given and 

required=128-22 .... 2-107988 
Then we have 1146 + 128 = 1272 = radius of a 4° 30' curve. 
Then say: 1146 : 1272:: 800 : 888 = arc AC. 

This case is equally applicable to changing the last radius used 
in a compound curve terminating in a parallel tangent. 



Locating Side Tracks, Etc. 



377 



PROPOSITION III. 

In case the preceding method should consume too much of the tangent 
C G, it is required to change the origin of the curve, also the length 
of radius, so that the required tangent may commence opposite to B, 
running parallel to B H, 



-%3. 




In this case the radiating point will be changed from M towards 
A and B, the radius shortened, and the point A moved towards K. 

Let the required distance between tangents, the given radius, 
and curvature be .as in Proposition II., then we have by logarithms: 
As the external secant of 40° . . 9-4848'79 

■ ;Is to radius 10-000000 

' 'So is 30 feet rr: 1-477 121 

■ To difference of radii = 98*23 . . 1 992242 



By natural external secants = - 



30 



•305407 



= 98- — 



And 1146—98 = 1048 = radius of a 5'' 28' curve. 
Then, as 1146 : 1048 :: 800 : 782 = length of 5° 28' curve. 
98 (natural tangent of 40°= '83910) = 82 feet. 
Produce tangent 82 feet from A to K, and curve from thence 732 
feet of a 5° 28' curve. 



a 2* 



V J 



3Y8 



Formula for Running Lines, 



PROPOSITION^ r\^. Fij. 3. 

Raving located a curve with a given radiu.% terminating in a given 
point, it is required to change the origin of the curve, also the 
radius, so as to pass through the same terminating point, with a 
different direction of tangent. 



FiS.3. 




.4 

Let the given radius MB equal 2292 feet; the given arc BD 
equal 1000 feet, containing 25" of curvature; the given tangents 
D F and D E make an angle of (say) 4°, DF being 400 feet, and EF 
= 28 feet 



28 



4° = angle E D F, consequently the angle 



9-098229 
8-971703 
3-360217 
3-233991 



"We have , ^ 

4 X 1-75 

OLD = 25= +4°= 29°. 
By logarithms : 

As versed sine 29° . 

Is to versed sine 25° 

So is radius given B M = 2292 

To radius required C L = 1714 feet. 
By tables 1714 feet = radius of 3° 20^' curve. 

PROPOSITION Y. Fig. 4. 

Having produced the two tangerUs to their intersection, it is required 
to connect them by a curve passing a given distance from the vertical 
point. 
Given the angle LCB = 31° 44', and C E = 50 feet, to find the 



'I 



Locating Side Tracks, Etc. 



379 



radius MA, By geometry, the angle A M E = | L C B = 1 5° 52'. 



IlgA, 



^y 



^ 



m 



By logarithms we have : 

As external secant 15° 52' = ^ L C B S-aOlTSO 

Is to 50 1-698970 

So is R. . . . . . . 10-000000 

To M A=1262=:R. of a 4° 32f curve 3-101181 

60 50 



By natural external secants 



ex. sec. 15° 62' '039603 



1262 ft. 



Case 2d. 

To find the tangent AC, or C B ; or point of curve. 

By logarithms : 

AsR 10 000000 

Is to AM =1262 .... 3-101181 
So is tangent 15° 62' ... 9-453668 

To AC = 388-8 .... 2-664849 

By natural tangents : 

1262 X (natural tangent 15° 52' = -26546) = 388 feet 
= C A = C B. 



380 



Formula for Running Lines, 



PROPOSITION VI. Fig. 5. 
Having located a curve connecting two tangents, it is required to move 
the middle of the curve any given distance, either towards or from 
the vertex. 

Let the angle TLG = 36° = whole amount of curvature; the 




arc A B C = 1 200 feet ; the radius A N = G N = 1910 feet, and I B 
= B F = 10 feet. 
It is required to find the radii H M and E O. 
We have by logarithms : 

External secant 18° = half of 36° = A N L S-YSYISS 

Is to 10 1-000000 

So isR 10000000 

To difference of radii = 183 feet . . "2^2847 

By natural external secants : -^rr 183 ft. 

•0546-95 

1910 + 183 = 2093 = MH = radius of a 2° 44' curve; 
and 1910 — 183 = 1727 = 0E= radius of a 3° 19' curve. 
By natural tangents : 
183 X (natural tangent 18° = -32429) = 69-4 = H A = AE. 



Locating Side Tracks, Etc. 



381 



PROPOSITION VIL Fig. 6. 
It is required to locate a tangent from an inaccessible point on a 

curve. 

Let A B C be the given curve with a R. of 1637 feet curving 3° 30' 
per 100 feet; C the inaccessible point. Assume a point B, if con- 
venient, at a given distance, say 300 feet, from C. Throw off a 
tangent, and measure, at right angles therefrom, B E = external 



Fig. 6, 




a 

secant of arc B C ; then to find by logarithms the distance B E, we 

have : 

As radius 10*000000 

Is to =: 1637 3-214122 

So is external secant 10° 30' == angle COB . 8231221 

To BE = 27-88 1-445343 

By natural external secants : 

1637 X (nat. ex. sec. 10° 30' = -017030) = 27*88. 
Measure the line B E =r 27 '88 feet at right angles to B II. Set 
the instrument over E, and turn off the angle BEC = 79° 30' = 
complement of 10° 30'. EOF will be the direction of the tangent 
required. 

Case 2d. 
Suppose there be no convenient accessible point between A and 
C, produce the curve to D, measure the external secant D F as 
before, place the instrument at F, and turn off the angle DFC. 
This will give the direction of the tangent F C as before. 

Case 3d. 
Should the lines A I and I C be more practicable for operating 



382 



Formulae for Running Lines, 



than the curve A B C, calculate and produce the tangent from A to 
I, the vertex of the curve ABC, and turn off the angle K I F = 
A O C, and make I C = A I, as calculated. 
Case 4th. 

Again, should the last method be found impracticable, and the 
chord AD clear from obstructions, measure the oho-rd AD, and 
turn off tangent from D. 

Suppose angle K AD = 25°, then we have 1037 x (nat. sine 25° 
= -42262) 2 = A D = 1384 feet. 

JVo^e.— The arc A B C D contains 50° curvature. 

PROPOSITION VIII. Fig. 7. 
It is required to find a curve which will connect two lines without pro- 
duci7ig the tangents to an intersection. 




The principle involved in this diagram affords an easy mode of solving a very interesting 
geographical problem. Suppose AE is a mountain near the sea or a very extensive 
level. Measure with an instrument for taking vertical angles the depression or " dip '' 
ot the horizon K E B = B O H ; then external secant K E B x radius of earth = A E — 
height of mountain. 



Locating Side Tuacks, Etc. 



Let the line be either a curve L A, H A> or a tangent D A, as the 
case may be. Suppose it impracticable, by reason of buildings or 
other obstructionsj to produce the tangent to a vertex x. 

At A lay off with the instrument a right angle to tangent, and 
produce it till it meets FB produced in E. Measure this distance, 
and the angle AEB; then its complement A OB will be the 
amount of curvature required to curve on to the tangent B F. 

Suppose the angle A E B = 65°, then A O B = 25°. Let A E be 
=^ 120 feet, then we have by logarithms: 

As external secant 25° ... 9*014427 

Is to 120 feet 2079181 

So is R . 10-000000 

To OB = 1160'8 = a4° 56^ curve . 3 '0647 54 
And 1160-8 x (tangent 25° = -46631) = EB = 541*28 feet. 
Then will be 25° of curvature ~- 4° 56^' = the rate of curvature, 
give the length of curve between the two given points A and B = 
506-2 feet. 

PROPOSITION" IX. Fig. 8. 

To draw a tangent to two curves already located. 

Let the curve CRAGH, of 2000 feet radius, be located from 

tangent CK; and let ESBD be a curve of 2605 feet radius, 

located from tangent E F. We are required to find the points A 

and B having a tangent common to both. 

Suppose R to be the point in the first curve, and S the point in 
the second. There being obstructions in the way, we will run the 
zigzag line R L P S, making R L tangent to R, and P S tangei.t 
toS. 

Suppose R L Q — 20° 
and TPS=:15°; 

let R L r= 1100 feet, L P = 1300, and P S = 1400. 
Assume radius N R as a meridian ; that is, suppose N R to be 
due north. Then will R L be due west, LP south 70° west, PS 
south 85° west, and radius S M north 5° west. These artificial 
courses, then, will show the relative bearings, with which we obtain 
the following traverse : 



Course. 


Distance. 


Northinff. 


Southing. 


Easting. 


Westing. 


North . . 
West . . 

S. 70° W. 
S. 85° W. 
N. 5° W. 


2000 
1100 
1300 
1400 
2605 


2000 
0000 
0000 
0000 
2595 07 


0000 

0000 
444-63 
12202 

0000- 


0000 
0000 
0000 
0000 
0000 


0000 
1100 
1221-60 
1394-66 
227 05 






4595-07 


566-65 


0000 


3943-31 



^84 



FoRMUL.« FOR Running Lines, 



then 



Difference northing and southing (4595*07 — 566-65) = 4028*42 ; 
8943*31 



4028*42 



= -97882 — natural tangent R N G = 44'' 2o' — course 




of NM=:K 44^ 23' west, and angle SMD = 39-23, or 44° 
23' — 6^ 



Locating Side Tracks, Etc. 



3g5 



To calculate MN make the difference of latitude 4028*42 = cosine 
44° 23', and the required distance N M =r radius. Then we have 

' . 4028-42 4028-42 

by natural cosines 



cosine 44*" 23' 



•71468 



= 5636-7 =MK 



Or bv logarithms : 

As cosine R N M = 44° 23' . . . 9-854109 

Is to R 10-000000 

So is difference of latitude 4028*42 . 3605134 

To N M = 5636-7 .... ""3^1025 

The triangles A N I and B M I being similar, we have by loga- 
rithms (Da vies' Legendre, book II., prop. X) — that is, by "compo- 
sition and division:" 

As :N^M = 5636-7 8-751025 

Is to R 10-000000 

So is sum of radii -4605 = (2000 + 2605| . 3*663230 

Tocosine ANI = BMIr=:35° IS' . . 9-912205 
Having now determined the angle R I^ I = 44° 23', and the 
angle A N I = 35° 13', the angle R N A becomes = to their differ- 
ence = 9° 10'". 

Therefore continue the curve from R towards A, 9° 10' of curva- 
ture, and we have the tangent point A required. Again, we have 
S M I = 39° 23', and the angle B M I == 35^ 13', consequently curve 
from S to B 4° 10' of curvature, and we have the tangent point B 
required. 

Now to find the length of tangent AB, multipl}^ the sum of the 
radii 4605 by the natural tangent of 35° 13', and we have the 
length required. 

Case 2d. Fig 




FonMULyE ¥on Running Lines, 



Suppose the two curves to be connected by a common tangent, 
instead of running in opposite directions as in Case 1st, curve the 
same Avay, as GHS and CD EL. It is required to find the 
position of the tangent S D. 

Assume the points H and E ; from H lay off tangent H I ; from 
E lay off tangent E F ; join F and I by a straight line, if convenient, 
or by a traverse, if there be obstructions. Let A II be an artificial 
meridian, and, as in Case 1st, calculate the distance A B, alto its 
course ~ angle HAG; this will give also the angle E B A. 

Suppose radius AH =1432-5, tangent H I = 500 feet, angle 
M I F rr 6°, I F = 1000 feet, N F T = 8°, E F = 600 feet, and radius 
E B = 2865 feet. We will then liave the following traverse, by 
which to find the course and distance of A B ; 



Course. 


Distance. 


Northing. 


Southing. 


Easting. 


Westing. 


ISTorth . . 
East . . . 

S. 84° E. . 
S.76°E. 
S. 14° W. 


1432-5 

500 
1000 

600 
2865 


1432-50 


104-50 
145-20 

2780-07 


500 

984-60 
582-20 


692-72 




Total 


1432-50 


3029-77 


2066-80 


692-72 



Difference of latitude = 1597-27 
Departure _ 1374-08 



diff. lat. 



1597-27 



departure = 1374-08. 
86026 = natural tangent 40° 42' 



course A B = angle H A G^ 

Diff. lat 1597-27 1597 27 



cosine course 



cosine 40*' 42' 



•75813 



:2106-86=distance 



AB. 

Then 



diff. radii 1432-50 



AB 



2106-86 



•67992 = natural cosine 47° 10' = 



D B A = S A G. 

Now 47° 10' —40° 42' = 6° 28' = H AS. Then curve from H 
6° 28' = 162 feet nearly to S. Now A B makes with B E an angle 
of 40° 42' + 8° + 6'' = 54° 42' Hence we must curve from E to D 
54° 42' —47° 10' = 7° 32' curvature = 377 feet distance. The 
points S and D will be the termini of the required tangent. 

Then difference of radii x natural tangent (D B E = 47° 10')=: 
1432-5 X 1-07864 = 1545 15 = AK = SD = length of tangent. 
Now when the two curves are so situated as to be seen the one 
from the other, assume two points as near as you can judge to the 
true termini of common tangent. Cause about a dozen small 



liOCATiNG Side Tracks, Etc. 



387 



straight stakes or pins to be set np endway about twentj^ feet 
apart from one of the assumed points or curves. Then set the 
instrument at the other, and see how tangent from instrument 
strikes the row of stakes. Note the difference, and move the instru- 
ment until tangent therefrom strikes as tangent to the row of 
stakes. Make a point where it does. Set the instrument over said 
point, and in like manner see how tangent from instrument strikes 
the other curve. Thus ^e dispense with all the previous calcula- 
tion. 



PROPOSITION X. Fig. 10. 

Having located two curves connected hy a tangent^ as in Case 2d, Prop. 
IJC., it is required to throw out the tdngent, and ijitrodace instead 
a curve with given radius. 

Let the radius AS = 1482o feet, B D = X6B7 feet, and their 
common tangent S D ::= 220 feet. It is required to find on the two 




curves two tangent points, X and Y, from which, if the required 
radius (say 2865 feet) be drawn, it will pass through the points A 
and B, intersecting in the centre P, oqui-distant from X and Y. 
Now in the triangle B A K we have given, difference of radti 



388 Formula for Running Links, 



BK=: 1637 — 1432-5=: 204-5; also, A K = S D = 220, to find the, 
angle K A B, its complement K B A = S A G,* and the distance A B. 

Then — —^ = -92954 = natural tangent of 42° 54^ r= K A B. 

Therefore its complement KB A = SAG = 47° 5^'. Now BK x 
secant KB A = 204*5 x 1-468801 = 300*37 = A B ; call it 300 
feet. Again, in the triangle B A P we have A B = 300, A P = 
2865 - 1432-5 = 1432-5, B P = 2865 — 1637 =r 1228. To find the 
angles A B P, B P A, and BAP, make A P = 1432-5 feet the base, 
and let Q be the foot of the perpendicular from B. Then bj trigo- 
nometry we have : 

AP; BP + BA ::BP — BA : PQ — QA, or 
1432-5 : 1228 + 300 :: 1228 — 300 : 989-8 = PQ — QA. Then 

1432-5 + 989-8 ^^ mn ir . 1432-5 — 989*8 ^, ^^, „^ 
= PQ =r 1211-15, and =:QA= 221-35. 

2 2i 

AO 2*^1*35 
Then — ^ = "^ ^ • = '73783 = nat. cos. of B A P =: 42° 27'. 
AB 300 

PO 1 211 *1 ^ 
Again, — -— = i^i— — = 98628 = nat. cos. of B P A = 9° 30'. 
^ ' PA 1228 

Is^ow YB P being a straight line, the angle YB A = 42° 27' + 
9° 30' = 51° 57' and X AP being a straight line, the angle X A G = 
BAP = 42° 27'. Now the angle SAG being 47° Sf the angle 
S A X will equal 47° 5^—42° 27' = 4° 38f , and Y B D = 51° 57' 
— 47° 5^' =4°51f. 

We therefore move back from S 4° 38^' of curvature, or 116 feet 
to X; also from D 4° 51|' of curvature, or 139 feet to Y ; we then 
have the points X and Y, which are to be connected with a 2° 
curve of 2865 feet radius. 

PROPOSITION XL Fig. 11. 

Having located a compound curve terminating in a given tangent^ it is 
required to change the p. c. c, also the length of the last radius, so as 
to pass through the same termi7iating point with a given difference 
in the direction of the tangent. 

Let the given curve H A be a 2° of 2865 feet radius compounded 
to A B, a 2° 30' curve 2292 feet radius, 800 feet in length, and con- 
taining 20° of curvature ; it is required to move the p. c. c. forward 
from A towards B, curving therefrom with a shorter radius than 
2292 feet, passing through the fixed point B on to tangent with 
2"^ 30' additional curvature. 

The following method, though not perfectly accurate, will be 

* Because the three angles In the triangle KAB =180°. Also the sum of the 
angles on one side the line B G = 180^ Subtracting from 180° the angle A tod the 
right angle at K, we have left the angle at B. Subtracting from 180° the angle 
A (as before) and the right, angle 8 A K, we have the angle SAG; hence the -angle 
KB A = the angle SAG. 



Locating Side Tracks, Etc. 



389 



found sufficiently so for most practical purposes. Had the 2° 
curve H A been continued 800 feet farther, to a point C, the varia- 
tion B C would be equal 28 feet.* Now by compounding to a 
2° 30' curve 1 turn off with the instrument for the chord AB 2° 



^ 
1 






1 






^ 




Mgdi 


\ 


\js 


^# / / 



20° 16° 

more than I would for the chord AC; for = 2° ; but if 

the instrument set at the required point P, with a backsight on A, 

20° 4- 2° 30' 
and a foresight on B, I turn off = 11° 15', that is 3° 15' 

instrumental deflection over and above that required for a con- 
tinuous 2° curve to C ; the curve P B will therefore be shorter than 
A B in the ratio of 3° 15' to 2° ; hence the proportion : 
31 : 2 :: 800 : 492 = length of curve P B. 
AP then will equal 800—492 = 308 feet of 2° curve; but 308 
feet of a 2° curve gives 6° 10' of curvature; hence PB contains 
22° 30' — 6° 10' = 16° 20' of curvature in 492 feet distance; then 

we have ^^-^-^ = 3-3198° = 3° 19', or 1728 feet radius for the 

4 9 2 

curve P B. It will be sufficiently accurate, however, to continue 
the 2° curve 310 feet to P, and then run 490 feet of a 3° 20' curve. 

Were H A a tangent by making A P the same lengtli and rate of 
curvature as above, the curve P T> would bo the same also. 

*2^ X 1.T5 X 8:^28. 



88* 



S90 Formula for Running Lines, 



PROPOSITIOIN' XII. 

Having located a compound curve terminating in a tangent, it is 
required to change the point of compound curvature so that the 
ctcrve will tenninate in a tangent parallel to a given tangent at any 
required distance perpendicular thereto. 

Rule. Divide the required distance between parallel tangents by 
the difference of radii of the two last branches of curve. From the 
cosine of total amount of curvature in last branch subtract this 
quotient; the remainder will be the natural cosine of amount of 
curvature required for last radius. 

Given a curve GOO feet long, 2805 feet radius, compounded with a 
curve of 1910 feet radius 400 feet long, then tangent; required to 
fix point of compounding, to give parallel tangent 30 feet outside 
or inside of tangent given : 

30 

-= -03141 

955 

400 feet curvature =12^ cosine 12° = '97815 

less -03141 



cosine of curvature required 18° 47' — -94674 

18° 47' — lii° — 6° 47' curvature to be used in moving p. c. c. back 
to throw a tangent 30 feet inside. 

If we move tangent inside 6° 47' -^ 2° = 339 feet. Length of 2° 
curve = 600 — 339 = 261. Length of 3° curve = 400 4- 226 = 626 ft. 

The entire length of curve by alteration becomes 261 + 626 = 887 
instead of 1000 feet as before, admitting of more tangent at the end. 

This last rule is applicable when the movement of the p. c. c. is 
retrograde or from the terminating tangent, thereby increasing the 
amount of curvature in last curve, and diminishing that of the pre- 
ceding curve. 

When it is required to move the point c. c. forward, diminishing 
the amount of curvature in last curve, ac^ the quotient of the required 
distance divided by difference of radii, to the cosine of given amount 
of curvature ; and the sum will be the cosine of the amount of curva- 
ture required in the last curve. Find the distance as before, and 
move the point forward the difference of curvature, always reckon- 
ing said difference according to the rate of curvature hack of p. c. c. 

PROPOSmOi^ XIIL Fig. 12. 

Having located a curve hetweefi two tangent points, it is proposed to 
length eji the radii at the two termini, and sJiorten the radius in the 
middle. 

Let the proposed curve be one of 1146 feet radius = 5°, 800 feet 
in length, and containing 40° of curvature. It is proposed to intro- 
duce at each end 100 feet of a 2° 30' curve = 2292 feet radium. 
Required the other radius. 

From the t. p. to the centre is 400 feet, or 20° of curvature. 



Locating Side Tracks, Etc. 



391 



Introducing 100 feet of a 2° 30' 2292 feet K, there will be 2° 30' 
of 2292 feet radius + 17° 30' of a shorter radius. 
By logarithms : 

As sine 17° 30' .... 
Is to 2'' 30' ..... 
So is diff. radii = 1146 feet = M . 
To difference between given radius 
and required = 167 = L . 



9-478142 
8-639680 
3-059185 

2-220723 



Fi^J2. 



yi^ 




3f> 



Given radius being 114(5, radius required will be 1146 — 167 =r 
979==LC=:a 5^51' curve. 



392 



FoRMULiC FOR RUXXING LiNES, 



PROPOSITION XIV. Lemma. Fig. 18. 
To divide a given angle into two parts, so that the tangents of the 
angles mil be in a given ratio. 

Let the required ratio be as three to five, and the given ande 
AD C = 30° ; let the straight line ABC be = 8. Make A C a 
chord of 60°, or twice 30°. Describe the circle A CD passing 




Jj Jit 



through A and C. At B, a distance of three from A, erect the per- 
pendicular B D, produce it to L, then ADB and BDC will be the 
angles required. For B D is common to two right angled triangles, 
and hence the tangents of the vertical angles are as AB to B C. 

To calculate the required vertical angles let A D B = a:, B D C = 
y, then A E L = 2 a?, and L E C == 2 j^ = central angle. Then AE C 
= 2{x-^ y) = 60°. 

Erect a perpendicular from E upon H, then will A E H = H E C 
= x + y. Then LEM (H being produced to M) equals x -\- y — 
^x = y — x; thenEL:=EC = R., andLM=BH. 'Then 
HC:(LM = BH) :: sine (a; + y) ; sine (^ — a:). 



Locating Side Tracks, Etc 



39,3 



But HC is half of AC, and B H is half BC — AB, therefore 
(B C + AB) : (B C — A B) :: sine {x 4- y) : sine {y — x\ that is, as 
the sum of the numbers expressing the ratio is to difference, so is 
sine of the given angle to sine of the difference required. 

By logarithms: 

0-903090 
0-301030 



As 3 + 6 == 8 
Is to 5 — 3 = 
So is sine SO" 
To sine y 



a; ==7° 10' 38" 



y-h x = 30°, y-^x = 1° 10' 



9-698970 
9-096910 



38", 



therefore 2y = 3riO' 38", y =z W 35' 19", x - W 24' 41". 

PROPOSITION XV. Fig. 14. 
From two fixed points^ having produced tangents uniting in a vertex 
at unequal distances from tkem^ it is required to locate a compound 
curve. 

Suppose the tangents produced tp E, and let A E = 606'1 feet, 



J' 



Jl^M. 






<>f 



7V 



394 Formulae for Running Lines, 



EB = 6206 feet, the angle FE A = 40°. Required tlie radii of a 
c. c. to join A and B, and also the point of compound curvature. 

We observe the external secant E C is common to both curves^ 
]S"ow by construction of the tables we have : external secant a = 
tangent a x tangent \ a, radius being unity. The angles EBC 
and EA C are measured by half their arcs C B and C A. 

40® 
Call these angles x and y respectively. Then x -^ yz=L — ^^ — = 

20^^ ; then 620*6 X tangent a- = 505-7 x tangent y, or 620-6 : 505*^ 
= tangent y : tangent x. Then by previous proposition 

620-6 + 505-7 : 620'6— 505*7 :: sine (a; ■\' y = 20°) : sine (a: — y) 
or, 1126-3 : 114-9 :: sine 20° i y — x. 

IS'either of the radii being given, we will assume the condition, 
that the p. c. C shall be in line with the vertex E and the centres O 
and D. We have by logarithms: 

As 1126-3 S-051654 

Is to 114-9 . . i v^ h^kiho%i\ tJa 2-060320 
So is sine 20° . . . . . 9-5 34052 
To sine (x— y) = 2° , . . . 8-542718 
Kow 0? + y = 20. Then ar = 9° + 2° = 1 T; consequently COB 
= 18°, and A D C = 22°. Now we have the length of tangent and 
curvature given, to find the radius. 
By logarithms : 

As tangent 18° . . . . . 9-511776 

Is to 620 6 2-792812 

So is R 10-000000 

To OB = 1910 feet . . . . 3-281036 
To find AD: 

As tangent 22° 9*606410 

Is to 505-7 . . . . . . 2-703895 

So is R 10-000000 

To AD = 1251-6 = 4^34' t. . . 3-097485 

1910 (external secant 18° = -051462) = 1251*6 x (external secant 

22° = -078535) = C E = 98*2 feet. 

PROPOSITION XYL Fig. 15. 

Let B he a point in a eiirve whose radius B F is piven^ and let D he 
another fixed tangent point. It is required to find point of c. €., the 
curve A B being produced, from which to start a curve to terminate 
in tangent at J)^ also the radius of last curve. 

Given the angles MDB, M B D, distance B D, and radius B F. 
Imagine the simple curve B C L to be run with a given radius B F 
till L N becomes parallel to D M. Now by the nature of a curve. 



I 



Locating Sidk Tracks, Exa 



395 



upon whatever point on the curve the transit be placed, the differ- 
ence between backsight on B and foresight on I, is alwa^'s the 




same, namely, -. Xovv^ at the true point of c. curvature 0, 

the difference between backsight on B and foresight on D is also 
B + D 



equal to - 



2 



-, therefore the transit reading the same on D as on 



L, OLD must be in the same straight line. 

Hence whenever the nature of the ground will admit of it, ecect 
a flagstaff at D, curve round from B towards L until taking a back- 
sight the foresight necessary to fall upon L should strike the flagstaff, 
at D, The transit will then be at the point of c. curvature sought. ! 

Then measure C D, and make this proportion ; sine II C L* : ^ CD 
:: R:a:=rOD. 

Suppose H C L = 8°, and the distance C D = 600 feet. Then by 
substituting in the above proportion, we have by logarithms : 



As sine 8° = H C L 
Is to 4- C D = 300 
So is k 
Toa-r= OD = 2855-0 



9-143555 

2-477121 

10-000000 

TS3356G 



* Because nCL -| COD. 



396 



FoRMULiB FOR RuNNING LiNES, 



"When the ground will not admit of this method, ascertain by 
measurement or calculation the distance from B to D. 

2 (B F) X — ti? - B L. Now angle L B D = -^ . The tri- 
angle will then have the two sides B D and B L, and the included 
angle B, to find the angle L D B =: CD B.f 

Kow in the triangle B C D we have the angle B C D = to the sup- 
B 4- T) 
piemen t of , also the angle at D, consequently the angle at B. 

These angles, together with base B D, determine the chord C D ; 
from which, with the angle HCL, calculate R as before. HCL 
becomes known from the fact, that C B D gives C B M = G C B, 

B 4- T) 
C B M being B — C B D. This taken from — ~ — will give the 

angle HCL required. 

PROPOSITION XVII. Fig. 16. 
Betweeti tivo tangents to locate a curve passing through a given point. \ 
Suppose A B and G D to be the tangents permanently fixed with 





B 4- D 

* Because N B L (isosceles) = ^ exterior angles at N and M = — x — , B being = 

NBD,andB-?^=?^=LBD. 
tLDB = (CLB = — T — ) — L B D which will make all the angles known. 



I 



Locating Side Tracks, Etc. 39Y 



reference to some agreement between individuals ; and let F be the 
given point at which it is necessarj^ to keep a given distance from 
some building or other object. JSuppose AB and CD produced to 
meet in E. The angle O E D, and consequently its half E B D, are 
known. The distance I E is also known. 

Let the angle O E D = 60°, let I F r= 17*5 feet. It is required to 
find the point B, so that the angle FBI shall = 30°. 

By natural sines : 

— — -— = 35 rzr FB =H D. 

sme 30 



Now 4/(35 + 17-5) X (35 — 17-5)* == V b'l-o x 17-5 = 30*3 == IB. 
Suppose I E measures 462 feet. Then B E will equal 462 + 30*3 
= 492-3. 
By similar triangles F B : B E :: B I : B K, or 

35 : 492 3 :: 30*3 : 426-2 = B K = D K. 
Then B D = 852-4 and B H == 852-4—35 = 817-4. 



Xow we have by geometry VBH x BF = BA, or 4/817-3 x 35 

rzr 169-1 =:B A. 

Hence A B + B E ^z: AE, or 169-1 + 492*3 = 661*4. 
To find radius : 

AE 661-4 

: r = rr IHo'O — 11. 

tangent 30 0-57755 

'Now suppose it is inexpedient to produce the tangents to a vertex, 
the angle O ED being known, find the point B as before, and turn 
off EBD=^OED, measure B D, and calculate by trigonometry 
the side E D = B E, and also B A as before. 

Again, suppose the angle at E is not known, neither is it prac- 
ticable to measure a direct line between the two tangents, calculate 
by traverse the true course and distance between any two con- 
venient points on the tangents by Proposition A^IIL, from which 
calculate the position of E. 

Without ascertaining the distance to E, the radius A G can be 
calculated thus : 

A F^ . ^ , . ^ , 200- 4000 

— = AG, or let AF = 200, then ^^- = -—=1146=: AG. 

Therefore commence at A, and run 800 feet of a 5° curve to G. 

PROPOSITION XVin. Fig. 17. 

Given the length of a common tangent D G r=z a^ and the angles of 
intersection /i and ni, to deter wine the common radius G E =z C JP" 
= radius of a reversed curve to unite the tangents H D and B L. 
Now D C = R X tangent | n, and 

C G = R X tangent ^ m ; we have therefore 
D G tangent =: 800 ft., n =: 16° and m = 12°. 

* The sum of two quantities multiplied by their difference is equal to the dif- 
ference of their squares. 

■ "34 ' 



898 



Formula for Running Lines, 



I M 




R== 



800 



800 



tan. -|-n + tan. \m 

32o6'7 X 
8256-7 X 



tan. 8°+tan. 6° -14054 + -10510 

•14054 = D C = I) A = 457-69 
-10510 = C G = G B = 342-27 

799-96 



= 3256'' 



Suppose it to be required to introduce 200 feet of tangent 
between the curves, that portion of the tangent D G taken by the 
two curves will be 600 feet. Then we have : 

800 : 600 :: 3256-7 : 2442*5 radius : 

800 : 600:: 457*69 : 348*27 ) _ , 

800 : 600:: 342*27 : 256*70 j ~ ^^^ langent. 

599*97 = " 



I 



ON REVERSED CURVES, TUR]S"OUTS, ETC, Fig. 18, 

A F =r 98 feet, A D = 102 feet, and D E =: 102 feet. 
Let G = gauge of track, and R = radius of turnout, x = distance 





:b / 







^'^^^^ 


\ . ^ . — 




* 


:p2 


w 






J 


u 



Locating Side Tracks, Etc. 399 



on chord from A, the origin of curve, to F, the point of frog ; then 
will ^ 

Now suppose R = SOO feet, and G = 6 feet, then will 

X — ^2 X 80U X 6 =: ^9600 = 98 feet nearly. 
Or let X = distance on main track to a point opposite of the 
froc:. Then will 



x= ^^G(2K — G) or 4^(2x800) — 0) = Vo x 1594 — 
V9564 r=: 97 •'7 9 feet. 
Hence the following rule is sufficiently correct for all practical 
purposes : 

Multiply tivice the radius by the gauge of track, extract the square root 
of tlie product, and we have the distance from origin of curve to 
jjoint of frog. 
Formula for angle of frog : G -t- R = versedrsine of curvature to 

froff = ancrle of froff. Ex. • — = -0075 = 7° 2'. 

Make the movable end of the switch rail such a distance from 
the origin of the curve, that the departure of a curve of that radius 
for that distance will be equal to the opening of that rail at the 
movable end, say 5|- inches. 

AVith an 800 feet radius, the distance from origin of curve to 

27 X 27 11 

openincr of switch rail will be = 27 feet, for -^^ — — = 54- 

^ '' ' 1600 24 ^ 

inches nearly. 

It will appear therefore that the opening of a 20 feet rail, with 
an 800 feet radius curve commencing at the other end, will be only 

^ . , . 20 X 20 ^ . , 

3 inches, for =3 inches. 

1600 

If we consider the movable rail as a movable tangent, and the 
origin of the curve as the opening of the rail, the angle of frog and 
length of curve will be obtained by Proposition XII. 

Example. 
A 20 feet rail, with 5^ inches opening, makes an angle with the 
main track = 1° 18', then on 6 feet gauge the distance from opening 
to other side = 5 feet 6^ inches ^554 feet. Then by Proposition 
XII. we have : 

cosine 1° 8' = *99974 



554 
800 



= -00692 

•99282 = cosine 6° 62' 
== angle of frog. 



400 



FORMUL.E FOR RuNNING LiXES, 



And 6° 52' — 1° 18' = 5" 34' = amount of curvature between open- 
ing of rail and point of fi-og. • 

By the fii*st method, when the distance between tracks = 13 
feet we have ^^13 x 800 = 102 feet nearly for distance from origin 
of curve to point of reversion- 

But if the point of reversion be made at the point of frog, the 
distance between nearest rails of tracks being 7 feet, we have 6 : 7 
:: 800: 933*3= radius of curve with which to leave frog, and 
6:7 :: 98 : 114"o = distance from frog to end of turnout. 

Or making the movable rail tangent, and its opening oi- inclies, 

angle of opening being 1' IS', the point of reversion being made at 

frog, to find the angle of frog, we have : 

cosine T IS' = '99974 



6-54 
933-3 



= -OOTSO 

•99274 == cosine 6'' 55' nearlv the same as before. 



TURXOUTS OX CURYES. Fig. 19. 

Suppose the turnout is on a curve running in the same direction, 
say a 2^ with a radius of 2865 feet. Xow an 800 feet radius gives 
a 7° 10' curve, and 7" 10' — 2" = 5"^ 10' = relative departure from 
main track But the radius of a 5^ 10' = 1109 feet ; then 



.V2 X 1109 X 6 = ar==115-3 — distance from origin of curve to 
point of frog. 
Therefore to make a turnout from a 2~ curve and running the 
same way would require 115 feet. 




Locating Side Tracks, Etc. 401 



If it were required to keep the distance the same as on a straight 
line, it would be necessary to make the 7° 10' curve a 9° 10' 
curve of 625 feet radius. 

If the 2° curve run in the opposite direction of the turnout, and 
the radius was 800 feet, then the convergence will be 7° 10' + 2° 
= 9° 10' curve, and the radius of a 9° 10' curve being 625 feet, we 
have : 



a: = 1^2 X 625 X 6 = '^'ToOO = 86'6 — distance from origin 
of curve to point of frog. 

When the main track is a curve^ and it is required to get on to a side 

track running parallel thereto. 

Note.— In treating of turnouts, When the main and side track are curves, the movable 
rail is considered a part of the curve used for turnoai, according to method 1st. 

Let E M be the main track on a curve of 2865 feet radius. It is 
proposed with a turnout from E, with a curve of 800 feet radius, to 
fall upon the side track B N, distant 13 feet ii^om the main track, 
and running parallel thereto. Now 2865 feet radius denotes a 2° 
curve, and 800 feet radius is a 7° 10' curve. Therefore the diver- 
gence of the curve E F from the curve E M is equal to (7° 10' — 2°) 
= 5° 10' curve; and the radius of a 5° 10' curve being 1109 feet, 
the divergence of the curve EF from the curve E M is equal to that 
of a curve of 1109 feet radius. 

By similar reasoning, the convergence of the curve FB towards 
being parallel with EM is 9° 10' per hundred feet, which may be 
expressed by a radius of 625 feet from tangent. Then we have 
1109+ 625 = 1734 : 1109 :: 13 : 8*31 = distance of point of rever- 
sion from main track. Now since x= '^'2E.G, we have by 
substituting %/ -J x 1109 x 8*31 = 135*7 = distance from origin of 
curve to point of reversion, radius used being 800 feet. The radius 
of relative curvature being expressed in the formula, we have the 
proportion 1109 : 625 :: 135*7 : 76*56 distance from reversion to 
2d track. 

Suppose it be required to put the side track on the opposite side, 
then we have 1734 : 625 :: 13 : 4*68 = distance of point of rever- 
sion from side track. Then we have the formula \^2 x 625 x 4*68 
= 76*48 distance from origin of curve to point of reversion. Then 
625 : 1109 :: 76*48 : 135*7 = distance from point of reversion to 
side track. 



ON RUNNING CURVES BY OFFSETS, OR WITHOUT THE 

USE OF AN INSTRUMENT FOR MEASURING ANGLES. 

Fig. 20. 

From a tangent E A let it be required to run a curve A B C D, 
having for its radius O C. To do this we have only to find H C 
and its half M C r= G B. 



34* 



402 



FoRMULvE FOR RuNNING LiNES, 




Suppose the chords A B, B C, C D are equal in length, being 100 
feet each. The chords, and consequently the arcs, being equal, the 
angle H B C is twice the angle GAB. But G A B is measured by 
half the arc A B — B C, consequently the angle H B C is measured 
by the whole arc B C. But the angle B C is also measured by 
the arc B C, consequently the angles H B C and B O C are equal 
IS'ow triangle B C is isosceles, and B H being equal to B C triangle 
II BC is isosceles also; consequently the two triangles are similar, 
and we have the proportion : 

B C^ 
H C : B C :: B C : B 0, consequently H C = ^.p^^r, or H C = 



BO* 



10000 



R 



Therefore M C = G B : 



AB" 
2R 



hence the following rule : 



The square of the uniform length of chord divided by radius will 
give the linear deflection from chord produced to curve, or half of 
this will give the deflection from tangent produced to curve. 

Examples. 



Suppose A O r= 2500 feet, then 
= 2 feet. 



10000 
"T500~ 



== H C = 4 feet, and G B 



Locating Sidk Tuaoks, Etc. 403 



Suppose A = 28G5 feet, the radius of a 2° curve, then we have 

II C =r J^— = 3-49 or 3-5 feet nearly ; and G B=^ of 3-5=r:l-75. 

2865 -^ 

Since the angle G AB = I'' the deflection for 1^ per hundred 

1'75 

feet is I'lOy or 0° 1' = ■ = '029, and one minute for one foot = 

60 

•00029, as by tables of natural sines. 

Case 2d. 

Suppose we run the curve around to a point which we will call 
station 10, or 1000 feet from beginning. The point Q, which is less 
than 100 feet distant from station 10, say 50 feet, being at station 
10 + 50. 

Suppose this a 2° curve compounded at station 10 + ^0 to a 3° 
curve of 1910 feet radius. Now the instrument setting on station 
10 with a backsight on station 9, the instrumental deflection to 10 
-j-50, 150 feet, will be 1° 30'. Now since 1° per 100 feet is 1-75, 
that of 1° 30' will be 2*62 feet. But the last chord being but 50 
feet, or half of a hundred, the deflection will be half of 2*62 = 1-31 ; 
hence we have the following rule : 

Multiply togetlier half the curvature in degrees = instrumental 
deflection between the backsight and point required, the length of 
the last chord and 1-75, and the product is the distance from chord 
produced to point required. 

Case Zcl 

Suppose the curve from 10+ 50 to station 11 is a 3° curve of 
1910 feet radius. Now the deflection from chord to tangent, from 
station 10 to station 10 + 50, is 0° 30', and the deflection from 
tangent to chord between 10 + 50 and 11 is 0° 46', therefore the 
entire deflection— 30' + 45' = 1° 15'. Now V 15' in a hundred 
= 1-75 X li = 2-18, and for 50 feet will be — 1-09 feet. 

Find station 12 by Case 2d, thus 2j;° (= instrumental deflection 
for 150 feet) x 1*75 = 3 '93 -^ deflection from chord produced to 
station 12 on curve. 

Continue the curve around as at first, observing to measure from 
curve to tangent the same deflection as from tangent to curve, or 
half the usual chord deflection ; the tangent point being supposed 
a full station. If not a full station, ascertain the tangent point by 
Case 2d, and the next full station on tangent by Case 3d. 

Having produced two tangents to an intersection at Cy it is required to 
comiect iliein with a curve of given length. Fig. 21. 

When the angle made by tangents is not greater than 15° the 
distance from vertex to the two ends of the curve will not differ 
materially from half the length of the curve. 



404 



Formula for Running Lines, 



Jig,2L 



je. 



Suppose the tangent B C produced 100 feet to E, measure CX =r 
100 feet, measure E X. Now suppose it is 21 feet. 

Kow the deflection of 1° for 100 feet is I'YS, and ■^— = 12° cur- 

1*7 5 

vature. 

Suppose it is required to divide the curve into 6 stations. Then 

21 

— = 3-5, the deflection for 2° in 100 feet. Hence it is a 2° curve. 
6 

Or 12° divided by 6 stations gives a 2° curve also. The deflec- 
tion being r=: 1"75 from tangent to curve. 

Between two fixed points to supply the intermediate points hy ordinates 
from the chord. Fig. 22. 



Mg.2Z, 



^ 



S 7 ff J 4 3 Z i 

By what has been previously demonstrated, the middle ordinate 

4x4 

4 to 4 will be expressed by — . At 3 the deflection from tan- 



Locating Side Tracks, Etc. 405 



1x1 2x2 

gent rnn each way from 4 to curve is — at 2 it is ■ — -. 

Jj ±{i 2 xt j 

4x4 I 

Hence the ordinate 4 to 4 = — •. Or 2 R being a common j 

2 K 

denominator, its relative value may be expressed by 4 x 4. At 
points 3 and 5 on chord the distance will be (4 x 4) — (1 x 1) — 
3x5 = 15. At 6 and 2 == (4 X 4) — (2 x 2) ==: 2 x 6 = 12, At 
7 and 1 = (4 x 4) — (3 x 3) = 1 x 1=^1. 
The ordinates are as follows : 

2x6 = 12 

3 X 5 = 15 

4 X 4 = 16. 

Then we observe that the sum of the two factors is equal, namely 
the length of chord. Hence the following rule : 

Multiply together the two segments of the chord or distance, 
divide by twice the radius, and the result is the distance from chord 
to curve. 

Suppose for example the radius = 5000 feet, then at points 1 andV 

100 X 700 70000 ^ ^ , ^ 

we have — rT^T^TT; = -z-z — ~ — i leet = offset at station 1 from 

10000 10000 

end. 

For 2 and 6 '^J^ ^ 12 ^ 2d offset. 
10000 

, 300 X 500 , ,, , 

For 3 and 5 -— — , — 15 = 3d offset, 

10000 

400 X 400 

and the entire length beino: 8 stations • =r 16 = greatest 

^ ^ 10000 ^ 

or middle ordinate. 

Had it been a 1° curve of 5730 feet radius, the ordinates would 
have been : 

1 X 7 X -P= 6-12 

2 X 6 X -J = 10-50 

3 X 5 X I =r 13-12 

4 X 4 X I = 14-00=middle ordinate; and 
so in proportion to any otlier rate of curvature in degrees. 

Hence when the rate of curvature is in degrees and no minutes, 
we have the following rule : 

Multiply together the distances in stations each side of the point, 
and the rate of curvature, deduct from this product \ of itself, the 
remainder will be the ordinate required. 

* The deiKirturc in 100 ft. of a V curve from tangent being — -75 ^ 3 of a foot. 



406 Formulae for Running Lijtes, 



Case 2d. 
Suppose that between tlie points and 8 there occurs a point of 
c. c, lor instance at 3 or 5, the curves compound from a 5000 feet 
radius to a 4000 feet radius. 

300 X 300 

By 1st method = 11 '25 = distance from end of chord 

^ 8000 

1 oOO X 500 
to tangent run Irom p. c. c, and r= 2o = distance irom 

other end to said tangent. I 

Measure from ends of chords respectively 11*25 and 25 feet; oni 
this line, at a distance 300 feet from 11*25 offset, and 500 feet from 
25 feet offset, would be the point of compound curvature sought. 

Or imagine either curve produced to a point opposite the end of 
the other ; calculate by Proposition XL, and measure the distance 
between the two curves, then on the new chord find the p. c. c. as 
by simple curves. Thus: 

300 X 300 300 X 300 _ ^^ ^ 

8000 10000 ~ ^ ^^* 

Measure 2*25 from the old chord, and you have the direction of 
the new. Having found the p. c. c. calculate the offsets from each 
chord separately. 

The above rule for ordinates, although not perfectly accurate, 
considering the divisor always = 2 R, while it is variable, is suffi- 
ciently near for centres to grade by, when the chord subtends not 
more than 20° curvature. 

This rule will aleo apply to placing centre points between 
stations. Thus : 

On a chord of 100 feet, radius 1000 feet, let it be required to 
locate a point 30 feet from one end and 10 feet from the other. 

rru X. 30 X To , ^^ 

Then we have • — = 1*05. 

2000 



FOR SPRI^^GING RAILS. 
Let L = length of rail and R = length of radius. Then : 






2R 
R 



= spring in inches. 



Locating Side Tracks, Etc. 407 



' X 12 

— - — =: spring in eighths of an inch, 

: spring in sixteenths of an inch. 

Example. 
Let the rail be 20 feet long, and the radius 1200 feet. Then 
24 X 20^^ _ 9600 
1200"" ■" 1200 
Hence the rule : 



, 8 

— 16- 



24 times the square of the length of rail in feet divided by length 
of radius in feet, will give the spring in middle in sixteenths of an 
inch. 

To find the length of chord for any rate of curvature (less than 8") 
not specified in the Table of Chords (p 414.) 

Example. 

Let it be required to find the length of chord corresponding to 
800 feet of curve for a ^7° 10' curve. 

1° curve gives . 769-01 
7° 15' curve gives 766-79 
Difference . . ?22 
Then 15 : 10 :: 2-22 : 1-48, and 769 01 — 1-48 = 767-53; 
or 15: 5:: 2-22: 74, and 766-79 + 0*74 == 767-53. 
The result, as obtained by the table of sines, is 767*54, only 
y^Q- of a foot difference. 

That is, sine 28° 40' x radius 800 x 2 = 767-54. 
Suppose now it be required to find the length of chord corres- 
ponding to 950 feet of a 6° curve. 

900 feet gives length of chord . 867 '45 
1000 " " '' . 955-37 

Sum 1822-82 

Mean 950 911'41 

Now sine 28° 30' x radius 955-37 x 2 = length of chord = 911-7], 
being only j^ ^^ ^ ^^^^ difference, so that this table will be suffi 
cient for ordinary purposes. For common rates of curvature for a 
less distance, say 650 feet, the variations from the true length would 
be scarcely perceptible. 



403 



FORMIL.E FOR RlNNLXG LlXZ>, ifco. 



Problem. — L-t A and C br two fxfd iangcnt points, tlie positions I 
of irhose tangents are deterniined by the angles DA C = m =. 18", ; 
B C jE =^ n =z G^, and the perpendicular distance D C ^= p ^= A^^oi 
//.* Required the amount of cnrvature in the arc A B, its reversion 
B C ajid the laujh of the comnion radius £ =^ Jf B by which 
the arcs A B a?'.d B C '.irc described. 




Let m = nrit vers, sine DA C, and n = nat. vers, sine B C E. 

Let .r = nat vers, sine ( A O B — -rn) = (B M C — n). 

( >r curvature A B = jn -f ^, and onrvatiire BG = n -{- xA 

T ^ J , yi - V V. s. IS" -^ V. s. 6"" 
io nna x we have, .>• = == — 



0-048944 -h 0-005478 

o =0 02<211 =: nat. vers, sine 13^ 23' 48'. 

Therefore arc AB= IS' -f 13^ 2S' 48' = 81^ 23' 48" and 6 0=6** + 
13° 23 48' = 10' 23 48'. Then by principles from which Proposi- 
tion XII. is derived, to find O B = K. 

perpd. dist. D C = p 

^ ^ ^ - R, Or 



AVe have 



twice nat. vers, sine AB — iiat. vt^rs. sine (wi— «) 
/)=: 463-0 463-5 



nat v.s.31'23 4i X 2 — nat. v.. s. 12 
463-5 
0-270"9SS 
31° 24' . _. ., , ^ ,19^24' 



! Then - 



0-146420 X :::-0-021S52 
= lTlO-4 — OB = radius of a 3'' 21' curve. 

= arcBC 



— - — ; ffi ves 937 ft. = arc A B, and — — -— srives 579 ft = arc 
go 21 '^ 8' 21'^ 



* If D C cancot be measured, me.^ure A C i.nd cr.^cTilate D C. Thus if A C = i 
1500 ft, we have 15C>0 x sine IS- = 15iX) > 0-3}9*>2 = 463-53. 

t D G E being equal to A O B. A O B - //i = A L G = C L G. Therefore a? = 
nat, vers, sine A LG = 13= 23 4S . 



TABLE OF RADII AND TPIEIR LOGARITHMS. 

(the CUKYATUEE is SUBTEXDED IJY a CIIOED of 100 FEET.) 



409 





DEGPwEE. 




1 DEGEEE. 




2 DEGREES. 


M. 


i Rudiiis 
|l.,.n>,e. 


( Jjo'-jn-iiliiu 
1 Infinite. 


M. 


|R.H„.. 


1 .Lcanth.n. | 


M. 


1 Radius, 


Logarithms. 


1 




1 





5730 


3-758128 





2865 


8-457115 


1 


343775 


5-536274 


1 


5636 


8-750949 


1 


2841 


3-453511 


2 i 


171887 


5-235244 


2 


5545 


3-743888 


2 


2818 


8-449937 


3 


114592 


5-059153 


3 


5457 


3-786940 


3 


2795 


8-446391 


4 


85944 


4-934214 


4 


5372 


8-730106 


4 


2772 


3-442876 


5 


68755 


4-837304 


5 


5289 


3-723367 


5 


2750 


3-439387 


G 


57296 


4-758123 


6 


52G9 


3-716737 


6 


2729 


3-435928 


7 


49111 


4-691176 


7 


5131 


3-71C206 


7 


2707 


3-432493 


8 


42922 


4-633184 


8 


5056 


3-708772 


8 


2686 


8-429089 


9 


38197 


4-5S2031 


9 


4982 


8-697452 


9 


2665 


3-425708 


10 


34377 


4-536274 


10 


4911 


8-691183 


10 


2645 


3-422356 


11 


31252 


4-494881 


11 


4842 


3-685023 


11 


2624 


3-419028 


12 


28648 


4-457(93 


12 


4775 


3-678947 


12 


2605 


3-415727 


13 


26444 


4-422331 


13 


4709 


8-672958 


13 


2585 


8-412448 


14 


24555 


4-890146 


14 


4646 


8-667057 


14 


2566 


3-409197 


15 


22918 


4-360183 


15 


4584 


3-661220 


15 


2547 


3-405967 


16 


21486 


4-332154 


16 


4523 


3-655469 


16 


2528 


8-402763 


17 


2G222 


4-305825 


17 


4465 


3-649792 


^17 


2509 


3-399581 


18 


19:;99 


4-281002 


18 


4407 


3-644189 


18 


2491 


3-396424 


19 


18C98 


4-257520 


19 


4352 


8-638656 


19 


2473 


3-893288 


20 


17189 


4-235245 


20 


4297 


3-633194 


20 


2456 


8-890176 


21 


16370 


4-214055 


21 


4244 


3-627800 


21 


2488 


3-887085 


22 


15626 


4-193852 


22 


4192 


3-622470 


22 


2421 


3-384016 


23 


14947 


4-174546 


23 


4142 


3-617196 


23 


2404 


3-380968 


24 


14324 


4-156064 


24 


4093 


3-612005 


24 


2387 


3-877943 


25 


13751 


4-138334 


25 


4045 


3-606866 


25 


2371 


3-374937 


26 


13222 


4-121802 


26 


3997 


3-6C1787 


26 


2355 


3-371954 


27 


12732 


4-104910 


27 


3952 


8-596766 


27 


2839 


3-36S9S9 


28 


12278 


4-089117 


28 


8907 


3-591803 


28 


2323 


3-366046 


29 


11854 


4-073876 


29 


3863 


8-5SG896 


29 


2307 


3-863121 


30 


11459 


4-059154 


30 


3820 


3-582044 


30 


2292 


3-860217 


31 


11G90 


4-044912 


81 


3778 


3-577246 


31 


2277 


3-857831 


32 


10743 


4-031125 


82 


3737 


8-572499 


32 


2261 


8-354466 


33 


10417 


4-017760 


33 


3697 


8-5678C4 


83 


2247 


3-351618 


34 


10111 


4-004797 


84 


3657 


8-563160 


34 


2232 


3-348789 


35 


9822 


8-992206 


85 


8619 


3-558564 


35 


2218 


3-345978 


36 


9549 


3-979973 


36 


3581 


3-554011 


36 


2204 


8-343187 


37 


9291 


3-908072 


37 


8544 


8-549516 


37 


2190 


3-340411 


38 


9047 


3-956493 


38 


3508 


3-545963 


38 


2176 


3-337655 


39 


8815 


8-945209 


89 


3473 


3-540654 


89 


2162 


8-334915 


40 


8594 


8-934216 


40 


3488 


3-530289 


40 


2149 


3-332193 


41 


8385 


3-923490 


41 


3404 


3-531968 


41 


2135 


3-329487 


42 


8185 


3-913029 


42 


8870 


3-527690 


42 


2122 


3-326799 


43 


7995 


8-902806 


43 


8388 


8-523452 


43 


2109 


3-324126 


44 


7813 


8-892824 


44 


3306 


3-519257 


44 


2:96 


8-321471 


45 


7639 


3-883063 


45 


3274 


3-515100 


45 


2083 


3-318831 


46 


7473 


3-873519 


46 


3243 


8-51(,985 


46 


2071 


3-31 62C8 


47 


7314 


3-864179 


47 


3213 


3-506907 


47 


2059 


3-313600 


48 


7162 


3-855036 


48 


3183 


3-502868 


48 


2046 


3-311008 


49 


7016 


8-846081 


49 


3154 


3-498866 


49 


2034 


3-308430 


50 


6876 


8-887308 


50 


3125 


3-494900 


50 


2022 


3-305869 


51 


6741 


8-828708 


51 


3097 


8-49{!970 


51 


2010 


3-308323 


52 


6611 


8-820275 


52 


8069 


8-487075 


52 


1999 


3-300797 


53 


6486 


8-812002 


53 


3042 


3-483205 


53 


1987 


3-298274 


54 


6366 


8-803885 


54 


3016 


3-479389 


54 


1976 


3-295771 


55 


6250 


8-795915 


55 


2989 


3-475596 


55 


1965 


3-293283 


56 


6139 


3-788091 


56 


2964 


3-471836 


56 


1953 


3-290809 


57 


6021 


3-780403 


57 


2938 


3-468108 


57 


1942 


2-288349 


58 


5927 


3-772851 


58 


2913 


3-464413 


58 


1931 


3-285902 


59 


5827 


3-76r)426 


59 


2889 


3-460748 


59 


1921 


3-283470 


CO 


5780 


8-758123 


60 


2865 


8-457115 


60 


1910 


3-281061 



35 



410 




Table of Radit axd their Logarithms. 






1 
3 DEGREES. 




4 DEGPwEES. j 


5 DEGREES. 


M. 1 


Rudi.s. 


J.osariilmi. i 


M. 


Radius 


Lo-aritl.n. 


M. 


Hariius. 


Logaiithm [ 


' 


1910 


8-281051 





1438 


3156151 





1146 


8-059290 


1 


1900 


3.278646 


1 


1427 


8-154544 


1 


1142 


3-057845 


i 2 


1889 


8-276258 ; 


2 


1421 


3-1525-18 


2 


11S9 


8-056407 , 


8 


1879 


3-273875 ! 


8 


1415 


3-150758 


8 


1185 


8-055010 ! 


4 


1869 


8-271508 


4 


14(9 


3-14S975 


4 


1131 


3-058542 ■ 


5 


1858 


3-2(^9155 


5 


1408 


3-147100 


5 


1127 


3-05-2115 


6 


1848 


8-202814 


6 


1393 


3-145481 I 


6 


1124 


3-050696 


7 


1889 


8-264486 


7 


1892 


8-143670 1 


7 


1120 


8-049279 


8 


1829 


3-262170 


8 


1886 


3-141916 j 


8 


1116 


8-047808 , 


9 


1819 


3-259S67 ! 


9 


1381 


8-140170 i 


9 


1118 


3-046461 


10 


1810 


3-257576 1 


10 


1375 


3-138430 


10 


11(9 


3-045059 


11 


1800 


8-255297 ! 


11 


1870 


3-136697 


11 


1106 


3-043662 


12 


1791 


8-253029 j 


12 


1364 


8-134977 


12 


11(2 


3-042268 


13 


1781 


8-250771 ; 


13 


1359 


3-138251 


18 


1C99 


3-040879 


14 


1772 


8-248580 


14 


1854 


8-131539 1 


14 


1(95 


3-089495 


15 


1768 


3-246297 j 


15 


1848 


3-129883 1 


15 


1C92 


3-G88114 


16 


1754 


8-244077 


16 


1343 


8-128134 1 


16 


1088 


3-086740 


17 


1745 


3-241867 : 


17 


1338 


3-126441 i 


17 


1085 


8-085868 


18 


1736 


8-289669 '■ 


18 


1383 


3-124756 ! 


18 


1081 


8-084002 


19 


1723 


3-237481 


19 


1328 


3-128075 ! 


n 


1078 


8-082686 


20 


1719 


8-235805 


20 


1822 


8-121404 : 


20 


1075 


3-081281 


21 


1710 


3-288140 


21 


1317 


8-119787 1 


21 


1071 


3-029927 


22 


1702 


3-28C9S5 


22 


1312 


3-118078 i 


22 


1G6S 


3-028577 


23 


1694 


3-228841 


23 


1307 


3-116423 


23 


1065 


3-027230 


i 24 


1686 


3-226707 


24 


1302 


3-114778 


24 


1061 


3-025890 


! 25 


1677 


8-2245S4 


25 


1293 


3-113184 


25 


1058 


3-C24552 


; 26 


1669 


8-222479 


26 


1298 


3-111401 j 


26 


1055 


3-028219 : 


: 27 


1661 


8-220869 


27 


. 12SS 


8-l( 9871 i 


27 


1052 


3-021889 


■ 28 


1658 


8-21S-277 


28 


1283 


8-1CS249 : 


28 


1048 


3-020565 


1 29 


1645 


3-216128 


29 


1278 


S-1C6632 


29 


1045 


8-019243 


: 30 


1687 


3-214122 


30 


1273 


8-105022 


30 


1042 


3-017927 


i 31 


1680 


8-212060 


31 


1269 


8-103418 i 


31 


1C89 


8-016614 


! 82 


1622 


8-210007 


32 


1264 


8-1(1818 


82 


1086 


3-015805 


! 33 


1614 


3-207968 


; 88 


1260 


8-10G225 


33 


1083 


3-018999 


1 34 


1607 


8-205930 


34 


1255 


8-C9S688 


34 


1080 


3-012698 


: 35 


1599 


8-203906 


; 35 


1250 


3-C97C56 


85 


1027 


3-011400 


^ 36 


1592 


8-201892 


36 


1-246 


8-( 95481 


36 


1G£4 


3-010107 


1 37 


1584 


8-199891 


87 


1241 


3-C 93910 


87 


1021 


3-008817 


1 38 


1577 


8-197890 


38 


1237 


3-( 92374 


88 


1017 


8-007532 


1 89 


1570 


3-195903 


89 


1232 


S-C907SS 


39 


1014 


8-006-249 


! 40 


15e3 


8-198925 


40 


1228 


8-089236 


40 


1011 


3-004972 


1 41 


1556 


8-191957 


41 


1224 


S->;.S7()89 


41 


IOCS 


8-0C8G98 


' 42 


1549 


3-189996 


42 


1219 


3-086147 


42 


1CC6 


8-0C2427 ; 


43 


1542 


8-1SSC45 


43 


1215 


3-G84610 


48 


1003 


3-001159 ; 


i 44 


1535 


3-1S6103 


44 


1211 


3-088079 


44 


1000 


2-999897 


' 45 


1528 


3-184168 


45 


1207 


8-081553 


45 


996-9 


2-998686 


: 46 


1521 


3-182-244 


46 


1202 


3-080038 


46 


994-;} 


2-997881 


! 47 


1515 


8-180327 


47 


1193 


8-078518 


47 


991-1 


2-9961-28 


: 4S 


1508 


3-178419 


48 


1194 


8-C770C2 


48 


983-3 


2-994880 ' 


49 


1501 


3-176519 


49 


1190 


3-075503 


49 


985-4 


2-993684 


60 


1495 


3-174627 


50 


1186 


3-074005 


50 


9S2-6 


2-992898 


51 


14S9 


3-172742 


51 


1182 


3-072511 


51 


979-3 


2-991156 


52 


1482 


8-170868 


52 


1178 


• 3-071022 


52 


977-1 


2-9S9921 


53 


1476 


8-169C01 


53 


1174 


3-069587 


58 


974-8 


2-988690 


54 


1469 


8-167142 


54 


1170 


8-068059 


54 


971-5 


2-98746:3 


55 


1468 


8-165590 


55 


1166 


8-066584 


55 


968-7 


i 2-986199 


56 


1457 


8-163447 


56 


1162 


8-065116 


56 


966-1 


2-985018 


. 57 


1451 


8-161612 


i 57 


1158 


8-068648 


57 


963-4 


' 2-983801 


58 


1445 


1 3-159784 


i 58 


1154 


8-062194 


58 


960-7 


i 2-982587 


59 


1489 


i 3-157963 


59 


1150 


8-060738 


59 


653-0 


i 2-981877 


i 60 


1483 


3-156151 


60 


1146 


8-059290 


60 


955-4 


2-980170 







Table of Radii and 


TiiEirv Log 


ARITIIMS. 


411 


6 DEGEEE8. 


7 DEGEEES. | 


8 DEGEEES. 


M. 


Radiuc. 


Logarithm. 


1 

1 M. 


Had in . 


I.ogaiitlmi. 1 


M. 


Radiu.o. 


Logavi.hm. 





055-4 


2-930170 





819-0 


2-913-295 1 





716-3 


2-855335 


1 


952-7 


2-978967 


1 


817-1 


2-912266 


1 


715-3 


2-854488 


2 


950-1 


2-977766 


2 


815-1 


2-911234 


2 


713-8 


2-853583 


3 


947-5 


2-976569 


i 3 


813-2 


2-910208 


3 


712-3 


2-852634 


4 


944-9 


2-975375 


; 4 


811-3 


2-9 9183 1 


4 


710-9 


2-S517S7 


5 


942-3 


2-974186 


i ^ 


8:9-4 


2-9J8161 


5 


709-4 


2-350891 


6 


93'J-7 


2-972997 


i <5 


SJ7-5 


2-907142 


6 


707-9 


2-349999 


7 


937-2 


2-971314 


! ^ 


835-6 


2-906124 


7 


706-5 


2-849107 


8 


9J4-(5 


2-970633 


' 8 


8:'3-7 


2-905111 


8 


705-0 


2-843219 


9 


932-1 


2-9-0456 


9 


8J1-9 


2-904097 


9 


703-6 


2-847329 


10 


929-G 


2-9i332S2 


: 10 


800-J 


2-903:90 


10 


702-2 


2-846445 


11 


9ir-i 


2-937111 


11 


79i-l • 


2 •9:2032 


11 


700-7 


2-845562 


12 


924-0 


2-935948 


; 12 


796-3 


2-9.1076 


12 


699-3 


2-344679 


18 


922-1 


2-964773 


13 


794-5 


2-900073 


13 


697-9 


2-843799 


14 


919-6 


2-963616 


i 14 


792-6 


2-399073 


14 


693-5 


2-342921 


15 


917-2 


2-962458 


15 


790-8 


2-893075 


15 


695-1 


2-842044 


16 


914-S 


2-931303 


16 


739-0 


2-597078 


16 


693-7 


2-341109 


17 


912-3 


2-930150 


' 17 


737-2 


2-393085 


-17 


692-3 


2-340290 


13 


9J9-9 


2-959 JOl 


1 ^^ 


785-4 


2-395094 


18 


690-9 


2-839424 


19 


9j7-5 


2-957354 


\ 19 


733-6 


2-394103 


19 


689-5 


2-333554 


2J 


905-1 


2-953711 


! 2) 


781 -S 


2-393118 


20 


688-2 


2-837087 


21 


9J2-8 


2-955572 


21 


730-1 


2-3921;M 


21 


636-3 


2-336821 


22 


9J0-4 


2-954434 


22 


778-3 


2-891151 


22 


685-4 


2-335956 


2:j 


89S-0 


2-953300 


23 


776-6 


2-890171 


23 


684-1 


2-835093 


24 


895-7 


2-952168 


! 24 


774-8 


2-839193 


24 


682-7 


2-834232 


25 


89-3-4 


2-951040 


! 25 


773-1 


2-388218 


25 


651-4 


2-338373 


2(i 


891-1 


2-949915 


^ 26 


771-3 


2-837-244 


26 


080-0 


2-.332515 


27 


838-8 


2-943792 


; 27 


769-6 


2-836272 


27 


678-7 


2-831059 


28 


886-5 


2-947673 


i 28 


767-9 


2-335308 


28 


677-4 


2-330805 


29 


884-2 


2-946555 


i 29 


766-2 


2-334336 


29 


676-0 


2-329953 


sa 


88^-0 


2-945452 


1 30 


764-4 


2-333371 


30 


674-7 


2-329102 


31 


879-7 


2-944330 


: 31 


762-8 


2-33-2409 


81 


673-4 


2-823253 


32 


877-5 


2-943-223 


; 82 


761-1 


2-331445 


82 


672-1 


2-327405 


33 


875-2 


2-942116 


: 33 


759-4 


2-380490 


33 


670-7 


2-826560 


34 


873-0 


2-941015 


' 84 


757-8 


2-879534 


34 


669-4 


2-325715 


S5 


870-3 


2-939914 


35 


756-1 


2-878580 


35 


663-1 


2-824373 ] 


3i3 


86S-0 


2-938819 


86 


754-4 


2-377627 


36 


666-9 


2-324032 i 


37 


866-4 


2-937722 


87 


752-8 


2-376678 


37 


665-6 


2-323192 1 


38 


864-2 


2-936633 


^ 88 


751-2 


2-375730 


38 


664-3 


2-322355 j 


39 


862-1 


2-935543 


39 


749-5 


2-374788 


39 


668-0 


2-321519 1 


40 


859-9 


2-934459 


: 40 


747-9 


2-373840 ! 


40 


661-7 


2-820085 


41 


857-7 


2-933337 


41 


746-3 


2-872900 


41 


660-5 


2-319352 


42 


855-6 


2-932295 


' 42 


744-7 


2-871959 


42 


659-2 


2-819021 


43 


853-5 


2-9:Jl218 


43 


743-1 


2-371022 


48 


657-9 


2-818191 


44 


851-4 


2-930142 


44 


741-5 


2-870086 : 


44 


656-7 


2-317863 


45 


849-3 


2-929 J70 


; 45 


739-9 


2-369153 i 


45 


655-4 


2-31 653t 


46 


847-2 


2-923)00 


i 46 


738-3 


2-868221 ; 


46 


054-2 


2-315712 


47 


845-1 


2-926933 


' 47 


736-7 


2-867-291 ; 


47 


053-0 


2-31438S 


48 


843-1 


2-925367 


: 48 


735-1 


2-866363 


43 


051-7 


2-814003 ! 


49 


841-0 


2-924306 


' 49 


733-6 


2-365438 


49 


050-5 


2-313246 ! 


50 


839-0 


2-923747 


50 


732-0 


2-364514 i 


50 


049-3 


2-8124-23 


51 


836-9 


2-922691 


; 51 


730-5 


2-863593 i 


51 


04.S-1 


2-3110U 


52 


834-9 


2-921637 


52 


7-28-9 


2'362673 ' 


52 


046-3 


2-810796 


53 


832-9 


2-92 )5S5 . 


53 


727-4 


2-861756 ■ 


53 


645-6 


2-309932 


54 


830-9 


2-919536 


54 


7-25-3 


2-860840 ■ 


54 


044-4 


2-S091O9 


55 


82S-9 


2-913489 


55 


724-3 


2-859926 ' 


55 


043-2 


2-808353 


56 


826-9 


2-917446 


56 


7-22-8 


2-659014 ■ 


56 


042-0 


2-307594 


57 


824-9 


2-916408 


: 57 


721-3 


2-858104 


57 


040-8 


2-300741 


58 


8-22-9 


2-915365 


i 58 


719-8 


2-857196 


58 


039-6 


2-305935 


59 


821-0 


2-914327 


59 


71b-J 


2-356239 


59 


038-5 


2-805130 


CO 


819-0 


2-913-295 


00 


716-8 


2-855885 


00 


037-3 


2-304327 



412 



Table of Radii and their Looaritiims. 





9 DEGEEES. 




10 DEGREES. 


1 


11 DEGREES. 


M. 


RRdu.S. 


i 

1 Logarithm. 


M. 


Radu . 


Lnjarifhrp, 


i M. 


Radius. 


] Logarithm. 





637-3 


1 2-804327 





573-7 


2-758674 


i 


521-7 


2-717397 


1 


636-1 


2-803526 


1 


572-7 


2-757953 


1 


52G-9 


2-716742 


2 


684-9 


2-802724 


2 


571-8 


2-757232 


2 


520-1 


2-716087 


3 


638-S 


2-801926 


3 


570-8 


2-756514 


3 


519-3 


2-7154S4 


4 


632-6 


2-801128 


4 


569-9 


2-755796 


4 


518-5 


2-714781 


5 


631-4 


: 2-800a32 


5 


569-0 


2-755:79 


5 


517-8 


2-714130 


6 


630-3 


2-799538 


6 


568-0 


2-754364 


1 6 


517-0 


2-713479 


7 


629-1 


2-793745 


7 


567-i 


2-753650 


i 7 


516-2 


2-7128J30 


8 


62S-0 


2-r97953 


8 


566-2 


2-752937 


i 8 


515-4 


2-712181 


9 


620-8 


2-797163 


9 


565-2 


2-752225 


! 9 


514-7 


1 2-711533 


10 


625-7 


2-796374 


10 


564-3 


2-751514 


: 10 


513-9 


2-71C887 


11 


624-6 


2-795587 


11 


563-4 


2-750804 


! 11 


513-1 


2-710241 


12 


623-5 


2-794801 


12 


562-5 


2-750096 


12 


5124 


2-709596 


18 


622-3 


2-794017 


13 


561-6 


2-7493S9 


18 


511-6 


2-708953 


14 


621-2 


2-793234 


14 


560-6 


2-748683 


14 


51C-9 


2-708310 


15 


620-1 


2-792452 


15 


559-7 


2-747978 


1 15 


510-1 


2-707668 


16 


619-0 


2-791673 


16 


553-8 


2-747274 


i 1^ 


509-3 


2-707027 


17 


617-9 


2-79 JS94 


17 


557-9 


2-746572 


1 17 


508-6 


2-706387 


18 


616-8 


2-790117 


18 


557-0 


2-745870 


! 18 


507-9 


2-705748 


19 


615-7 


2-789340 


19 


556-1 


2-745170 


19 


507-1 


2-705110 


20 


614-6 


2-78S566 


20 


555-2 


2-744471 


20 


506-4 


2-704473 


21 


613-5 


2-787794 


21 


554-8 


2-743773 


21 


505-6 


2-703837 


22 


612-4 


2-787021 


22 


553-4 


2-743076 


1 22 


504-9 


2-7032G2 


23 


611-3 


2-786252 


28 


552-6 


2-742380 


1 28 


504-1 


2-702568 


24 


610-2 


2-785482 


24 


551-7 


2-741686 


i 24 


503-4 


2-701984 


25 


639-1 


2-784715 


25 


550-8 


2-740990 


1 25 


502-7 


2-701802 


26 


608-1 


2-783948 ! 


26 


519-9 


2-74030O 


26 


501-9 


2-700671 


27 


607-0 


2-783183 1 


27 


549-0 


2-7396G9 


27 


501-2 


2-700040 


28 


605-9 


2-782420 


28 


548-2 


2-738918 


28 


500-5 


2-699410 


29 


604-9 


2-781657 


29 


547-3 


2-738229 


29 


499-8 


2-698782 


30 


603-8 


2-780897 


30 


546-4 


2-737541 


30 


499-0 


2-698154 


31 


602-8 


2-780138 


31 


545-6 


2-736854 


31 


493-3 


2-697527 


32 


601-7 


2-779379 


32 


544-7 


2-736169 


32 


497-6 


2-696901 


33 


600-7 


2-778622 


33 


543-8 


2-735434 


38 


496-9 


2-696276 


34 


599-6 


2-777863 


34 


543-0 


2-734800 


34 


493-2 


2-695652 


35 


598-6 


2-777113 


35 


542-1 


2-7:34118 
2-733436 


35 


495-5 


2-695029 


36 


597-5 


2-776360 


36 


541-3 


36 


494-8 


2-694407 


37 


59a-5 


2-775608 


37 


540-4 


2-732756 ! 


37 


494-1 


2-693785 


38 


595-5 


2-774858 


38 


539-6 


2-782077 1 


88 


493-4 


2-693165 


39 


594-4 


2-774108 


39 


538-8 


2-731398 1 


39 


492-7 


2-692545 


40 


593-4 


2-773361 


40 


537-9 


2-730721 1 


40 


492-0 


2-691926 


41 


592-4 


2-77-2616 


41 


537-1 


2-730045 i 


41 


491-8 


2-691308 


42 


591-4 


2-771870 


42 


536-3 


2-729370 1 


42 


490-6 


2-690692 


43 


590-4 


2-771124 


43 


535-4 


2-728696 | 


43 


489-9 


2-690076 


44 


589-4 


2-770388 1 


44 


534-6 


2-728028 


44 


489-2 


2-6S9460 


45 


538-4 


2-769642 i 


45 


533-8 


2-727351 


45 


483-5 


2-688846 


46 


587-4 


2-768902 ; 


46 


532-9 


2-726684 


46 


487-3 


2-688233 


47 


586-4 


2-768163 


47 


532-1 


2-726010 i 


47 


487-1 


2-687620 


48 


585-4 


2-767426 


48 


531-8 


2-725342 


48 


486-4 


2-687008 


49 


584-4 


2-766689 


49 


530-5 


2-724674 


49 


485-7 


2-686398 


50 


583-4 


2-765955 | 


50 


529-7 


2-724008 


50 


485-0 


2 685788 


51 


582-4 


2-765223 


51 


523-9 


2-728342 


51 


484-4 


2-685179 


52 


581-4 


2-764489 


52 


5-28-0 


2-722677 


52 


488-7 


2-684570 


53 


580-4 


2-763758 


53 


527-2 


2-722014 ; 


58 


483-0 


2-683968 


54 


579-5 


2-763028 


54 


526-4 


2-721351 ! 


54 


482-3 


2-688357 


55 


578-5 


2-762299 


55 


525-6 


2-720690 


55 


481-7 


2-682751 


56 


577-5 


2-761572 


56 


524-8 


2-720019 


56 


481-0 


2-682146 


57 


576-6 


2-760845 


57 


524-0 


2-719370 


57 


480-3 


2-681542 


58 


575-6 


2-760120 
2-759898 


58 


528-2 


2-718711 


58 


479-7 


2-680939 


59 


574-6 


59 


522-5 


2-718054 


59 


479-0 


2-680337 


60 


573-7 


2-758674 


60 


521-7 


2-717897 

1 


60 


478-3 


2-679735 



Table of Radii and their Logarithms. 



413 



12 DEGREES. 


13 DEGREES. 


14 DEGREES. 


M. 


I Radius. 


Logaii.hm. 


M. 


! Ra.Uu.. 


! Lo?,'ari(lim. 


M, 


j Radius. 


j Lngariaim. 





478-8 


2-679785 


! ^ 


441-7 


2-645111 





410-3 


2-613075 


1 


477-7 


2-679185 


1 


441-1 


2-644557 


1 


409-8 


2-612.561 


2 


477-0 


2-678535 


2 


i 440-5 


2-644004 


2 


4(:9-3 


2-612048 


8 


476-3 


2-677936 


3 


I 440-0 


2-643451 


3 


408-3 


2-611535 


4 


475-6 


2-677288 


4 


1 4S9-4 


2-642900 


4 


4C8-3 


2-611G23 


5 


475-0 


2-676741 


5 


; 438-9 


2-642848 


5 


407-9 


2-610511 


6 


474-4 


2-676145 


6 


1 488-3 


2-641793 


6 


4C7-4 


2-610000 


7 


478-8 


2-675549 


7 


487-8 


2-641248 


7 


406-9 


2-6C9490 


8 


473-1 


2-674954 


8 


487-2 


2-64CC99 


8 


406-4 


2-6CS980 


9 


472-5 


2-674360 


9 


436-7 


2-640150 


9 


4C6-0 


2-608471 


10 


471-8 


2-673767 


10 


1 436-1 


2-6396C3 


10 


405-5 


2-607962 


11 


471-2 


2-673175 


11 


! 435-6 


2-689056 


11 


405-0 


2-607454 


12 


470-5 


2-672584 


12 


j 485-0 


2-688510 


12 


404-5 


2-606946 


13 


4G9-9 


2-671993 


13 


4-34-5 


2-637964 


13 


4C4-0 


2-6C64S9 


14 


469-2 


2-671408 


14 


433-9 


2-637419 


14 


4C3-6 


2-605933 


15 

16 


468-0 
468-0 


2-670814 
2-67C226 


15 
16 


488-4 

432-8 


2-63G875 


15 


403-1 
4G2-6 


2-605428 


2-636382 


^6 


2-604923 


17 


467-8 


2-669688 


17 


482-3 


2-0357S9 


17 


402-2 


2-604418 


^^ 


466-7 


2-069052 


18 


431-8 


2-635247 


18 


401-7 


2-60S914 


19 


466-1 


2-668466 


19 


431-2 


2-6347(.5 


19 


401-2 


2-603411 


20 


465-5 


2-667881 


20 


430-7 


2-634164 


20 


400-8 


2-6G29C8 


21 


464-8 


2-667-297 


21 


430-2 


2-633624 


21 


400-3 


2-6C24C6 


22 


464-2 


2-666713 


22 


429-6 


2-638CS5 


22 


399-9 


2-601905 


23 


468-6 


2-666131 


23 


429-1 


2-682546 


23 


899-4 


2-601404 


24 


468-0 


2-665549 


24 


428-6 


2-6S2308 


24 


398-9 


2-60C9C4 


25 


462-3 


2-664968 


25 


428-0 


2-631471 


25 


398-5 


2-600404 


26 


461-7 


2-664387 


26 


427-5 


2-63G934 


26 


398-0 


2-599905 


27 


461-1 


2-668808 


27 


427-0 


2-630888 


27 


397-6 


2-599406 


28 


460-5 


2-663229 


28 


426-4 


2-629863 


23 


397-1 


2-598908 


29 


459-9 


2-662651 


29 


425-9 


2-629328 


29 


896-7 


2-598411 


SO 


459-2 


2-662074 


30 


425-4 


2-628794 


30 


396-2 


2-597914 


31 


458-6 


2-661498 


31 


424-9 


2-628261 


31 


895-7 


2-597418 


32 


453-0 


2-660922 


32 


424-4 


2-627728 


32 


395-3 


2-596922 


33 


457-4 


2-660347 


33 


423-8 


2-627196 


33 


394-8 


2-596427 


34 


456-8 


2-659773 


34 


428-3 


2-626665 


34 


894-4 


2-59£933 


35 


456-2 


2-659200 


35 


422-8 


2-626134 


85 


398-9 


2-595489 


36 


455-6 


2-658628 


36 


422-3 


2-625604 


36 


393-5 


2-594946 


37 


455-0 


2-658056 


37 


421-8 


2-625074 


37 


398-0 


2-594453 


38 


454-4 


2-657485 


38 


421-3 


2-624546 


38 


892-6 


2-59S961 


39 


4o3-8 


2-656915 


39 


420-7 


2-624018 1 


39 


392-2 


2-5984G9 


40 


453-2 


2-656845 


40 


420-2 


2-623490 i 


40 


891-7 


2-592978 


41 


452-7 


2-655776 


41 


419-7 


2-622963 1 


41 


391-2 


2-592487 


42 


452-1 


2-655208 


42 


419-2 


2-622487 i 


42 


890-8 


2-591997 


43 


451-5 


2-654641 


43 


418-7 


2-621912 i 


43 


890-4 


2-591508 


44 


450-9 


2-654075 


44 


418-2 1 


2-621387 i 


44 


390-0 


2-591019 


45 


450-3 


2-658509 


45 


417-7 


2-620863 


45 


389-5 


2-590531 


46 


449-7 


2-652944 


46 


417-2 


2-620339 


46 


389-1 


2-590043 


47 


449-1 


2-652380 


47 


416-7 


2-619816 


47 


388-6 


2-589556 


48 


448-6 


2-651816 


43 


416-2 


2-619294 


48 


888-2 


2-589069 


49 


448-0 


2-651254 


49 


415-7 


2-618772 


49 


387-8 


2-588583 


50 


447-4 


2-650691 


50 


415-2 1 


2-618251 


50 


387-3 


2 -588(97 


51 


446-8 


2-650130 


51 


414-7 


2-617781 


51 


386-9 


2-587612 


52 


446-2 


2-649570 


52 


414-2 


2-617211 


52 


386-5 


2-587128 


53 


445-7 


2-649010 


53 


413-7 


2-616692 1 


53 


886-0 


2-586644 


54 


445-1 


2-648451 ! 


54 


413-2 


2-616173 


54 


385-6 


2-5S6161 


55 


444-5 


2-647892 i 


55 


412-7 1 


2-615655 


55 


885-2 


2-585678 


56 


444-0 


2-647335 | 


56 


412-2 


2-615188 


56 


384-8 


2-585196 


57 


443-4 


2-646778 j 


57 


411-7 


2-614622 


57 


884-3 


2-584714 


58 


442-8 


2-646222 1 


58 


411-2 


2-614106 


58 


383-9 


2-584233 


59 


442-2 


2-645666 


59 


410-8 


2-618590 


59 


883-5 


2-583752 


60 


441-7 


2-645111 


60 


410-3 


2-613075 


CO 


883-1 


2-588272 



35* 



41^ 



Table of Chords. 



TABLE 

Of Chords corresponding to every 100 feet on curve frcym 200 to 1000 
feet, calculated to every 15 minutes^ rate of curvature, from 15 
minutes to 8 degrees, radius of\° being b'lSOfeet. 



1 i 200 feet. 



15' 

80' 
45' 
1° 

1" 15 
1° 30' 

r45' 

2» I 
2° 15' I 
2° 30- : 
2^45' 

go 

3° 15' 

8° 30' 

I 3' 45 

! 4=15' 
! 4-30' 
I 4° 45' 
• 5" 
! 5° 15' 
I 5°30' 

5=45' 
I 6= 

6=15' 



300 feet. 400 feet. 500 feet. 600 feet. 



200-00 

200-00 

200 

199-99 

199-99 

199-93 

199-93 

199-97 

199-96 

199 95 

199-94 

199-93 

199-92 

199-91 

199-89 

199-83 

199-86 

199-S5 

199-33 

199-31 

199-79 

199-77 

199-75 

199-73 

199-70 

199-63 

199-65 

199-63 

199-60 

199-57 

199-54 

199-51 



300-00 

299-99 

299-93 

299-97 

299-95 

299-93 

299-91 

299-83 

299-35 

299-31 

299-77 : 

299-73 

299-63 

299-63 

299-57 

299-51 

299-45 

299-33 

299-31 j 

299-24 I 

299-16 

299-08 

299-00 

293-90 

293-81 

293-72 

293-61 

293-51 

293-40 

293-29 

298-17 

298-05 







400-00 


499-99 


399*98 


499-96 


899-95 


499-91 


399-92 


499-85 


399-88 


499-76 


399-83 


499-66 


899-77 


499-53 


899-70 


499-39 


899-61 


499-23 


899-52 


499-05 


399-42 


493-85 


899-32 


493-63 


399-19 


493-39 


899-07 


498-14 


893-98 


497-86 


893-73 


497-57 


893-68 


497-25 


893-46 


496-92 


398-28 


496-57 


893-10 


496-20 


897-90 


495-81 


897-70 


495-40 


897-49 


494-98 


397-26 


494-58 


397-03 


494-07 


396-80 


493-60 


396-54 


493-09 


396-28 


, 492-57 


396-01 


i 492-03 


395-73 


491-47 


495-44 


490-90 


395-14 


490-31 



599-98 
599-93 
599-84 
599-73 
599-58 
599-40 
599-18 
598-94 
598-65 
598-34 
597-99 
597-61 
597-19 
596-74 
596-26 
595-74 
595-20 
594-62 
594-00 
593-86 
592-68 
591-97 
591-22 
590^ 
589-64 
588-81 
587-93 
587-02 
586-08 
585-11 
584-12 
583-08 



700 feet. 800 feet. 



900 feet. 1000 ft 



699-97 
699-89 
699-76 : 
699-57 i 
699-33 I 
699-04 
698-69 j 
698-30 ! 
697-84 I 
697-84 I 
696-78 ' 
696-17 ' 
695-50 i 
694-79 
694-02 
693-20 I 
692-32 
691-40 
690-42 I 
639-39 i 
638-30 ' 
687-17 i 
685-93 i 
684-75 1 
683-46 
632-13 
680-73 
679-29 
677-79 
676-25 
674-66 
673-01 



799-96 
799-84 
799-64 
799-86 
799-00 
798-56 
798-04 
797-44 
796-76 
796-01 
795-lT 
794-25 
798-26 
792-18 
791-03 
789-80 
788-49 
787-11 
785-64 
784-10 
782-48 
780-79 
779-01 
777-16 
775-24 
773-26 
771-16 
769-01 
766-79 
764-49 
762-12 
769-67 



899-94 

899-77 
899-49 
899-09 
898-57 
897-95 
897-20 
896-35 
895-33 
894-80 
893-10 
891-80 



888-85 
887-21 
885-45 
883-58 
881-61 
879-52 
877-32 
875-02 
872-61 
870-08 
867-45 
8&4-72 
861-90 
853-98 
855-87 
852-72 
849-45 
846-09 
842-62 



999-92 

999-69 

999-30 

I 993-75 

I 998-05 

i 997-18 

! 996-15 

994-98 

993-65 

992-17 i 

990-52 

988-73 

986-78 

934-68 

982-42 

! 979-99 

I 977-46 

i 974-75 

' 971-89 

: 968-87 

965-72 

962-42 

958-96 

955-3T 

951-63 

947-75 

943-71 

939-54 

935-23 

930-78 

926-20 

921-4t 



Tables of Vkuskd. Sim-:s and Extekxal Secants. 415 



TABLES OF NATURAL AND LOGARITHMIC VERSED SINES, 
AND EXTERNAL SECANTS. 




On the Construction of the Tables of Versed Sines and External 
Secants. 

In tlie above figure it is required to find the value of versed sine 
F B = C G, of arc B C = A B angle a, and of external secant C D in 
terms of sine C F and tangent B I). 

The chord B C = 2 sine ^ B C, and angle F C B is measured hy 
I arc AB =^ arc BC. 

Therefore making chord B C radius, B F will be the sine of angle 
F CB, and we have: 

Versed sine B F = 2 x sine FC B =2 x (sine ^ a)l 

That is, twice the square of sine of half given arc = versed sine. 
Making C F radius. B F becomes tangent, and we liave, versed 
sine B F = CF x tangent F CB, or sine a x tangent ^ a. 

Now by similar triangles v. s. a : ex. sec. a :: cos. a : radius ; 

find V. s. a : ex. sec. a :: sine a : tangent a; 

or, ex. sec. a =: y. s. a x radius ) , . . _^„. i ^ 

' - J- = tan. a x tangent | a. 

cosine a ) 
Then log. v. s.a = log. sine a + log. tan. | a — (10 = log. of R.), 
and log. ex. sec. a = log. v. s. « + 10 — log. cos. a. ; 
or, log. ex sec. a = log. tan. a + log. tan. ^ a — 10. 



Example. 



Log. sine 40°— 0-808067 
Log. tan. 20°= 9 -061066 
Log. V. s. 40°= 9-369133 



Log. tan. 40° = 9-92381 3 
Log. tan. 20° = 9-561066 
Ex. sec. 40° = 9-484879 



il6 






^ 


^ERSED Sines. 










DEGEEE. 


1) 
1 DEGEEE. i 


1 
2 DEGEEES. 


Min. 


Nat No 


I-ogarilhm. 1 


Min. 


Nat. No. 


Logaritl.m. 


Min. 


Nat No 1 


■Lo-aii^hm. 





0-000000 


0-000000 1 





0-000152 


6-132714 i 





0-0006(9 


6-734740 : 


1 


•000000 


2-621422 


1 


-000157 ; 


•197071 i 


1 i 


•000619 


-791948 


2 


•000000 


3-223482 : 


2 ; 


•0001G2 


•211194 i 


2 


•000630 


•799 97 


8 


•000000 


•580664 ; 


8 


•000168 


•225091 ; 


^ i 


•000640 


•806187 


4 


•000000 


•830542 


4 


•000173 


•238770 i 


4 


•000650 


•813-219 


5 


•000001 


4-024362 


5 


•000179 


•252227 i 


.5 


•C0C661 


•820194 , 


6 i 


•000002 


•182724 i 


6 


•000184 


•265496 i 


6 


•000672 


-827114 


7 


•000002 


•316618 ' 


7 


•000190 


•278557 i 


7 


•000632 


•838980 ; 


8 


•000003 


•432602 ' 


8 


•000196 


•291426 ! 


8 


•000693 


•840792 ' 


9 


•000003 


•534906 ' 


9 


•000201 


•304106 ' 


9 


•0OC704 


•847551 ' 


10 


•J00004 


•626422 i 


10 


•000207 


-316604 i 


10 


•000715 1 


•854257 


11 


•000005 


•7C9200 ' 


11 


•000213 


•328925 1 


11 


•000726 i 


•86C912 


12 


•000006 


•784784 \ 


12 


•000219 


•841072 ' 


12 


•000737 ; 


-867516 


13 
14 


•000007 
•000008 


•854306 i 
•918678 ' 


11 


•000225 
•000232 


•853051 . 


13 


•C00748 , 


-874070 

-330577 


•864868 ' 


14 


•0007^9 1 


15 


•000010 


-978602 


15 


•000238 


•876528 ; 


15 


•000771 , 


-887033 


16 


•000-011 


5-034662 


16 


•000244 


•888032 ; 


16 


•000782 . 


-893444 


17 


•000012 


•087316 


17 


•000251 


•899387 ; 


17 


•000794 


•399306 


13 


•000014 


•13G966 i 


18 


•00C257 


•410592 ; 


18 


•00CS05 


-906128 


19 


•000015 


•183924 ! 


19 


•000264 


•421657 i 


19 


•00C817 


-912398 


20 


•000017 


•228480 1 


20 


•000271 


•432582 ! 


20 


•000829 


•918618 


21 


•000018 


•270856 ! 


21 


•00C278 


•443372 


21 


•000841 


-924800 


22 


•000020 


•311266 : 


22 


•000234 


•454030 ' 


22 


•000853 


•93C937 


28 


•000022 


•849377 


28 


•000291 


•464588 


23 


•000365 


•987C32 


24 


•000024 


•386842 


24 


•00C298 


•474960 


24 


•000377 


•943084 


25 


•000026 


•422302 ' 


25 


•000306 


•485238 i 


25 


•0008S9 


•949093 


26 


•000029 


•456366 


26 


•000318 


•495396 


26 


•000902 


-955062 


27 


•000031 


•4S9140 , 


27 


•000320 


•505488 I 


27 


•000914 


•960991 


28 


•000C33 


•520736 


28 


•000328 


•515364 ! 


28 


•00C926 


•966879 


29 


•000035 


•551216 


29 


•000385 


•525179 i 


29 


•000939 


•972726 


30 


•000088 


•580662 


80 


•000843 


•534882 i 


80 


•00C952 


•978536 


31 


•000040 


•6G9143 


81 


•000350 


•544480 j 


81 


•000964 


•9S43C5 


82 


•000043 


•636720 : 


82 


•000358 


•553972 


32 


•00C977 


-990088 


38 


•000046 


•668449 


83 


•000366 


•568362 


33 


•000990 


•995783 


34 


•000049 


•639376 I 


84 


•000874 


•572651 


34 


•001013 


7-001391 


35 


•C00052 


•714558 


35 


•000382 


•581841 


35 


•001016 


•007013 


36 


•000055 


•739024 . 


36 


•000390 


•59C987 i 


86 


•001029 


•012597 


87 


•000058 


•762821 i 


37 


•000898 


•599936 ' 


37 


•001C43 


•018147 


88 


•000061 


•785984 


38 


•000406 


•608845 1 


88 


•001056 


•023660 


89 


•000064 


•808549 


: 89 


•000415 


•617662 


89 


•001069 


•029139 


40 


•00006S 


•330538 


i 40 


•000423 


•626393 : 


40 


•001083 


•034584 


41 


•000071 


-351935 


1 41 


•000431 


•685034 1 


41 


•001097 


•039995 


42 


•000075 


•872916 


\ 42 


•000440 


•643591 ! 


42 


•001110 


-045372 


43 


•00007-8 


-893358 


1 43 


•000449 


•652064 ; 


43 


•001124 


-050717 


44 


•000082 


-913322 


44 


•000458 


•660456 


44 


•001138 


•056027 


45 


•000086 


■ -932345 


' 45 


•000466 


•668768 ■ 


45 


•001152 


•061307 


46 


•000090 


' -951932 


; 46 


•000475 


•676999 


46 


•001166 


•066554 


47 


•000C93 


' -970611 


47 


•000484 


•685156 


47 


•001130 


•071771 


48 


•000C97 


•933893 


48 


•000493 


•693234 


48 


•001194 


•076955 


49 


•000102 


6-006770 


! 49 


•000503 


•701240 


49 


•001 2C8 


! •082119 


50 


•000106 


-024354 


■ 50 


•000512 


•7C9171 


50 


•001222 


•087282 


51 


•000110 


' -041559 


1 51 


•000521 


•717082 


51 


•001237 


•092325 


52 


•000114 


•058420 


i 52 


•000531 


•724820 


52 


•001251 


•C.97388 


58 


•000119 


•074965 


; 53 


•000540 


•732540 


53 


•001266 


•102423 


54 


•000123 


•C91200 


i 54 


•000550 


•740192 


54 


•001231 


-107428 


55 


•000123 


-107146 


55 


•000559 


•747778 


55 


•001295 


-112405 


i 56 


•000133 


: -122738 


.: 56 


•000569 


-755297 


56 


■ -001310 


•117353 


i 57 


•000137 


j -138167 


i »7 


•000579 


•762752 


57 


1 -001325 


•122272 


i 58 


•000142 


! -153268 


1 58 


•000589 


•770144 


58 


! -001340 


•127165 


59 


-000147 


•163116 


59 


•000599 


•777472 


59 


•001355 


•132031 


60 


•000152 


•182714 


60 


•000609 


-784740 


60 


•001370 


•136868 









External S 


ECANTS. 






417 












DEGREE. 


1 DEGREE. 


2 DEGREES. 


Min. 


Nat. No, 


Logarithm. 


Mill. 


Nat No. 


Logarithm. 


Min. 


Nat. No. 


Logarithm 





0-000000 


•000000 





0^000152 


6-182780 





0-000610 


6-785005 


1 


•000000 


2-626422 


1 


-000157 


-197139 


1 


-00062Q 


-792217 


2 


•000000 


3-228482 


2 


•000163 


-211265 


2 


-000630 


•799371 


3 


•000000 


•580604 


3 


•000168 


-225164 


3 


•000040 


-800465 


4 


•000000 


•830542 


4 


•000173 


•238845 


4 


•000651 


-813502 


5 


•000000 


4-024362 


5 


•0G0179 


•252305 


5 


•0006G1 


-820481 


6 


•000001 


•182725 


6 


•000184 


•265576 


6 


•000672 


•827406 


T 


•000002 


•316619 


7 


•000190 


•278689 


7 


•000683 


-834276 


8 


•000003 


•482603 


8 


•0C0196 


•291511 


8 


•000(:94 


-841093 


9 


•000003 


•534907 


9 


•000201 


•804193 


9 


•000704 


•847857 


10 


•000004 


•626424 


10 


•000207 


-31G694 


10 


•000715 


•854568 


11 


•000005 


•709209 


11 


•000213 


•329018 


11 


•000726 


•861227 


12 


•000006 


•784787 1 


12 


•000219 


•841167 


12 


•000738 


•867836 


13 


•000007 


•854309 1 


13 


•000225 


•853149 


13 


•000749 


-874895 


14 


•000008 


•918682 1 


14 


•000232 


•864969 


14 


•000700 


•88C907 


15 


•000010 


-978606 ! 


15 


•000238 


•876631 


15 


•000772 


-887368 


16 


•000011 


5-034667 


16 


•00C244 


•888188 


16 


•000783 


-893784 


17 


•000012 


•087821 


17 


•000251 


•899486 


^7 


-000795 


-900157 


18 


•C00014 


•136974 


18 


•000257 


-410704 


18 


•000806 


-906473 


19 


•000015 


•183933 


19 


•00C264 


•421772 


19 


•C00S18 


•912748 


20 


•000017 


•228487 


20 


•000271 


•432700 


20 


•000830 


•918978 


21 


•000018 


-270864 


21 


•000278 


•448493 


21 


•000842 


•925165 


22 


•000020 


•311275 


22 


•00G285 


•454154 


22 


•000854 


•9313C8 


23 


•000022 


•349882 


23 


•00C292 


•464685 


23 


•000866 


•9374C8 


24 


•000C24 


•886853 


24 


•C00299 


•475C90 


24 


•000878 


•943465 


25 


•000026 


•422314 


25 


•000306 


•485371 


25 


•000890 


•949479 


26 


•000029 


•456378 


26 


•000313 


•495532 


26 


•00C9C3 


-955454 


27 


•000081 


•489153 


27 


•000320 


•505577 


27 


•00C915 


-9613SS 


28 


•000083 


•520750 1 


28 


•000328 


•515506 


28 


•00C927 


•967282 


29 


•G00C36 


•551280 


29 


•000335 


•525325 


29 


•000940 


-973134 


80 


•000038 


•580679 


80 


•000343 


•535031 


80 


•00C953 


•978950 


81 


•000040 


•619151 


81 


•000350 


•544632 


81 


•00G965 


•984724 


82 


•000043 


•636739 


82 


•000358 


•554128 


32 


•000978 


-990463 


83 


•000046 


•663469 


83 


•000366 


•568521 


83 


•000991 


•996163 


84 


•000049 


•689397 


84 


•000374 


•572813 


34 


•001014 


7-001827 


35 


•000052 


•714581 j 


35 


•000382 


•582007 


85 


•001017 


•007455 


86 


•000055 


•739048 


36 


•000890 


•591106 


86 


•001031 


•013044 


37 


•000058 


-762846 


37 


•000398 


•6001C9 


87 


•001044 


•018600 


88 


•000061 


•786013 i 


88 


•000406 


•609022 


38 


•001057 


•024119 


89 


•000064 


•808577 


39 


•000415 


•617842 


89 


•001071 


•029604 


40 


•000068 


•880567 


40 


•000423 


•626577 


40 


•001084 


•035055 


41 


•000071 


-852016 


41 


•000432 


•685222 


41 


•001093 


•040471 


42 


•000075 


•872948 


42 


•000440 


•643:82 


42 


•001111 


•045854 


43 


•060078 


•893387 


43 


•000449 


•652259 


43 


•001125 


•051205 


44 


•000082 


•913358 


44 


•000458 


•660655 


44 


•001139 


•056521 


45 


•000086 


•932882 


45 


•000466 


-668771 


45 


•001153 


•061807 


46 


•000C90 


•951971 


46 


•000476 


•677206 


46 


•001167 


•067061 


47 


•000C94 


•970652 


47 


•000485 


•685366 


47 


•001181 


•072284 


48 


•000097 


•988940 


48 


•000494 


•693448 


48 


-001195 


•077474 


49 


•000102 


6-006814 


49 


•000503 


•701458 


49 


•001210 


•082644 


50 


•000106 


-024400 


50 


•000512 


•709393 


50 


•001224 


•087768 


51 


•000110 


-041607 


51 


•000522 


•717258 


51 


•001238 


•C 92862 


52 


•000114 


•058470 


52 


•000531 


•725051 


52 


•001253 


•( 97932 


53 


•000119 


•075017 


53 


•000540 


-732775 


53 


•001268 


•102978 


54 


•000123 


•C91254 


54 


•000550 


-740431 


54 


•001282 


•107985 


55 


•000128 


•107202 


55 


•000560 


•748021 


55 


•001297 


•112968 


56 


•000133 


•122846 


56 


-000570 


•755544 


56 


•001312 


-117922 


57 


•000137 


•128227 


57 


•000579 


•763004 


57 


•001327 


•122848 


68 


•000142 


1 •153330 


58 


•000589 


•770400 


58 


•001842 


•127747 


59 


•000147 


i •168180 


59 


•000599 


•777732 


59 


•001357 


•132620 


60 


•000152 


•182780 


60 


•000610 


•785005 


60 


•001372 


•137464 



41S 








Versed 


Sixes. 










3 DEGREES. 




4 DEGEEES. 




5 DEGEEES. 


Jlin. 


Ntt. No. i Logsri.hm. j 


Min. 


Nat No 


LosariiLm. , 


MLr. 


Na:. No. 


L.^.i:l.n,. 





CrOOlSTO ! 7 


136S68 i 





0-:X)24:36 


7^886669 ' 





0-003505 


7-58(^9 


1 


•C013S6 : 


141679 ! 


1 


•01.2456 


•890273 1 


1 


•0(>85S1 


•583272 


2 


•0;:14.:l ; 


146404 


2 


•C02477 


•393824 


2 


-0C3556 


•586156 


3 


•001417 i 


151225 ; 


3 


•002497 


•897455 1 


3 


•0035S2 


•5S9u26 


4 


•001432 : 


155955 


4 


•002518 


•401019 


4 


•00£9.7 


•591SS6 


5 


•C0144S : 


1G0661 


5 


•0C2538 


•404572 1 


5 


•X'S933 


•594735 1 


6 


•0014<3:3 


165:342 


6 


•0('2559 


•40S1C8 


6 


•0CS959 


•597573 ; 


7 


•G01479 


1699S9 


7 


•002580 


•411629 


7 


•003955 


•6«}0410 i 


s; 


•001-195 : 


174G29 


8 


•C026C1 


•415137 


8 


•004010 


•6032^ ! 


9. 


■C01511 ^ 


179236 


9 


•002622 


•418632 


9 


•004C37 


•6(;6048 ' 


10 


-001527 : 


1S3S19 


10 


•002643 


•422111 . 


10 


•aJ4063 


•6(;8S52 


11 


•00151^3 


1S5377 


11 


•002664 


■425577 


11 


•0C4CS9 


•611 G47 


13 


-CC1559 


192912 


12 


•C(.26So 


•429:29 


12 


•004116 


•614434 


13 


•OaOTG ; 


197422 


13 


•002707 


•43-2468 


13 


•004142 


•617-210 


14 


•a€a:92 i 


201910 


14 


•0G2728 


•435S92 


14 


•GC41G9 


•61995J 


15 


•OCrlGCS ; 


20G376 


15 


•002750 


•439303 


15 


•004195 


•622740 


16 


•G01C2O : 


21CS17 ; 


16 


•CC2771 \ 


•4427C2 


16 


•0142-22 


•6-25490 


IT 


•001041 : 


215237 


17 


-0C.2793 


•446087 : 


17 


'a;4'24S 


•6252:34 


13 , 


•GOlOoS 


219032 ; 


18 


•0C.2S15 


•449458 


18 


•004275 


•63(966 


19 


•OOlCTo i 


224018 ' 


19 


•002837 


•45231? 


19 


•0H)43C2 


■6:33G92 


20 


•C&1692 \ ' 


22S360 1 


20 


•002-S59 


•456162 '-^ 


20 


•004329 


•6:364C9 


21 


•0017v9 f 


232G92 


21 


•0C.2531 


•459494 1 
•4€2S15 ' 


21 


•004356 


•6:59117 


22 , 


•001726 


237000 


22 


•00-29G3 


22 


•0(4382 


■641516 


23 i 


•C01743 


2412S9 


23 


•002925 


•466121 


23 


•004410 


•6445<^6 


24 ' 


•C01760 


245555 


24 


•002947 


•469417 


•24 


•004433 


•647170 t 


25 


•001777 


249SC2 


25 


•002970 


•472C99 


25 


•004466 


•649864 


26 , 


•001795 


254r.26 


26 


•0(i2992 


•475969 i 


26 


•0C4493 


•652532 


2t 


•G01S13 ! 


25S232 


27 


•003015 


•479227 


27 


•004521 


•655190 


2S ! 


•001530 : 


262116 


28 


•008Ci37 


•482472 


28 


•0(4543 


•657840 i 


29 


•001 S47 


'2665S2 


29 


•0C8060 


•4857C5 


29 


•004576 


•660482 , 


30 


•001S65 


-270725 


30 


•008083 


•485926 


30 


•0(46(4 


•668116 ; 


31 


•C01S53 


274S52 


81 


•0031C6 


•492137 


31 


•004GS2 


•665743 I 


32 


•0019 1 


27S957 


32 


•003129 


•495834 


32 


•OC4660 


•668360 


33 


•C01919 


283043 


38 


•003152 


•49S5-28 


33 


•0C468S 


•67(972 L 


34 


•001987 


2S71C9 


34 


•003175 


•5(,1C94 


84 


-0C4716 


•678574 


35 


•0C1955 


■291156 


35 


•008198 


•504557 


35 


•0(4744 


•676168 : 


36 


•0C1973 ' 


2951S7 


86 


•0C8221 


•5080CS 


86 


•0(14773 


•678759 


37 


•001992 


299196 


87 


•C0824i. 


•511147 


37 


•004801 


■681334 


38 


•OG2010 


303190 


88 


•0C8-268 


•514275 


38 


'(K4S30 


•688906 


39 


•0G2G23 


307162 


89 


•003291 


•517S91 


89 


•CC4858 


•686470 


40 


-0C-2C47 ; 


311119 


40 


•008815 


•520498 


40 


•0(4857 


•68962^ P 


41 


■002066 


815C56 


41 


•003389 


•528593 


41 


•004916 


•691574 


42 


•C02CS5 


315977 


42 


1)08862 


•526677 


42 


-0(.4044 


•594116 ! 


43 


•002103 ' 


322579 


43 


•ooa3S6 


•529750 


43 


•0(4973 


•696649 


44 


•0C2122 


326764 


44 


•003410 


•532512 


44 


-005002 


•699176 


i 45 


•002141 


•33C634 


45 


•0C3434 


•535563 


45 


-00=5031 


•701696 


46 


•0C2160 


•3:M4S3 


46 


•003459 


•53S904 


46 


-005061 


•704208 : 


• 47 


•0C2179 


•83S316 


47 


•0034S8 


-541938 


47 


-0(5C'90 


•706713 


' 48 


•00219S : 


•34213:3 


48 


•008507 


•544953 


48 


-00-5119 


•709210 


1 40 


•002218 


•845932 


49 


•008531 


•547961 


49 


-005149 


•711700 ; 


i 50 


•C02237 


•349716 


50 


•00S556 


•55(961 


50 


-005178 


•714184 1 


51 


•002257 


•353452 


51 


•003551 


•553945 


51 


•005208 


•716658 1 


' 52 


•002276 


•357233 


52 


•0Ci3605 


•556927 


52 


•005238 


•719128 


53 


•0C2296 


•36C967 


5:3 


•003630 


-559595 


53 


•005267 


■7215S9 


i 54 


•0':2316 


•364657 


54 


•003655 


-562S52 


54 


-0(5297 


•7-240^ 


: 55 


•OC233(3 


•36S39J 


55 


• -003650 


•565800 


55 


-005S27 


•726492 


^ 56 


•CK}2355 


•37-2076 


56 


•003705 


•565737 


56 


•005S57 


•7-259:34 


57 


•0C2375 


•375746 


57 


•003730 


-571665 


57 


V.C5a37 


•731367 


! 5S 


•0C2396 


•379403 


55 


' -003755 


-574582 


58 


•G05417 


•733796 


1 59 


•002416 


•353043 


59 


' -008780 


•577492 


59 


•00'5448 


•736217 


1 60 

i 


•002436 , 


•356669 


60 

1 


1 -008805 


•580889 


60 


•005478 


•738630 

1 









External 


'^ lie ANTS. 






419 










8 DEGREES. 


j 


4 DEGREES. 


5 DEGPwEES. 


Mill. 


Naf. No, 


1 I . I thin. 


\ Mm. 


Nil . Xo 


i.c-^. It . 


Min. 


j Nat. No. 


Liirr r:th)n. 





0-Jijl3T2 


1 7-137464 


i 


0^0G2442 


7-337728 





0-OOS820 


7-582ri45 


1 


•ooiass 


•142281 


! 1 


•002462 


•;91346 


1 


•003845 


•584946 


2 


•J01403 


•147073 


2 


•002483 


•394951 ' 


2 


-003871 


•5879J4 


S 


•001419 


•151S41 


3 


•0C2533 


•893541 


3 


-003897 


•590715 


4 


•301434 


-156577 


4 


•0C2524 


•4j2114 1 


4 


•003923 


•593586 


5 


•001450 


•161290 


5 


-002545 


•405676 j 


5 


•003949 


•596446 


6 


•0(;i465 


•165978 


6 


•002566 


•409221 1 


6 


•003975 


•599301 


7 


•001481 


•170642 


1 7 


•002537 


•412751 • 


7 


-004001 


•602144 


8 


•001497 


•175279 


8 


•002608 


•416203 


8 


-004027 


•604979 


9 


•001513 


•179893 


9 


•002629 


•419772 


9 


•004053 


•607805 


10 


•001529 


-1844-83 


10 


•002650 


•4-23261 


10 


•004080 


•610620 i 


11 


•001545 


-189048 


11 


•002671 


•426736 1 


11 


•0C41O7 


•013427 


12 


•001562 


-193590 


12 


•002693 


•430197 


12 


•004133 


•016225 


13 


•001578 


•198107 


13 


•002714 


•433645 : 


13 


-004159 


•619013 


14 


•001594 


•202602 


14 


•002736 


•437079 


14 


-004186 


•621704 


15 


'001611 


•207075 


15 


•002757 


•440499 


15 


•004213 


•624506 


16 


•00162S 


•211523 


16 


•002779 


•443907 1 


16 


•004240 


•627327 


IT 


•001644 


•215951 


17 


-002301 


•447802 t 


17 


-004267 


•630188 1 


18 


•001061 


•220358 


18 


-002323 


•450682 


18 


•0C4294 


•682327 


19 


•001G78 


•224736 


19 


-0G2845 


•454051 


19 


-004321 


•635504 


20 


•001695 


•229095 


20 


•002367 


•457405 : 


20 


•004343 


•63S293 


21 


•001712 


•233435 


21 


•002389 


•460747 \ 


21 


-004375 


•641013 


22 


•001729 


•237750 


22 


•002911 


•464077 


22 


•004403 


•643724 


23 


•001746 


•242047 


23 


•002934 


•467893 


23 


•004480 


•646426 


24 


•001768 


-240320 


24 


•002956 


•470699 


24 


•00-M58 


•6491C2 


j 25 


•001731 


-250575 


•25 


•0C2978 


•473991 ' 


25 


•004485 


•651808 


26 


•001798 


'254306 


26 


•003001 


•477210 i 


26 


•004513 


•654488 


27 


•001816 


•259020 


27 


•008024 


•480538 1 


27 


•004541 


•657158 


28 


•001833 


•263211 


28 


•003046 


•483793 i 


28 


•004569 


•659320 


29 


•001851 


•2673S5 


29 


•003069 


-437036 ' 


29 


•004597 


•662474 


30 


•001869 


■271536 ' 


30 


•003392 


-49r;207 j 


80 


•004025 


•665120 


31 


•001837 


•275671 


31 


•003115 


•493438 1 


31 


-004653 


•667759 


32 


•0010D5 


•279783 i 


32 


•003133 


•496094 


82 


•004681 


•670383 


33 


•001923 


•233377 : 


33 


•003161 


■499394 


83 


•004710 


•673013 


34 


•001941 


•237951 ■ 


34 


•003185 


•503075 : 


34 


•004738 


•675627 


85 


•001959 


•292006 ! 


35 


•003208 


•506248 


35 


•004767 


•678233 


36 


•001977 


•29604-5 


36 


•003232 


•509409 : 


36 


•004793 


•680837 


37 


001996 


•330062 


37 


•008255 


•512558 ; 


37 


•004S24 


•683424 


38 


•002314 


•304064 i 


38 


•003279 


•515697 


38 


•004853 


•686009 


39 


•002032 


•808044 


39 


•003302 


•518323 : 


89 


-004882 


•6S8585 


40 


•OQ2051 


•3120U9 1 


40 


•003326 


•521940 ' 


40 


•004911 


•091154 


41 


•002070 


•315954 : 


41 


•003350 


•525045 '■ 


41 


•004940 


•693714 


42 


•002J89 


•319383 i 


42 


•003374 


•528140 ' 


42 


•004909 


•696269 


43 


•002108 


•323793 i 


43 


•003393 


•531223 : 


43 


-004993 


•G9S814 


44 


•002127 


•3276S8 ; 


44 


•003422 


-534296 : 


44 


•005.23 


•701354 


45 


•002146 


•331565 


45 


•008446 


•537357 i 


45 


•005057 


•703887 


46 


•002165 


•335422 


46 


•003471 


•540409 1 


46 


•005080 


•706411 


47 


•002184 


•339263 1 


47 


•003495 


•543448 i 


47 


•005116 


•708929 


48 


•002203 


•343U89 ; 


48 


•003519 


•546479 ; 


48 


•OOaUf) 


•711439 


49 


•002223 


•346896 


49 


•00354^4 


•549497 1 


49 


•005175 


•718942 


60 


•002242 


•350689 


50 


•003569 


•5525G8 ' 


50 


•005-J05 


•716439 


51 


•002262 


•354463 


51 


•003593 


•555506 I 


51 


•005235 


•71S926 


52 


•002281 


•358223 


52 


•003618 


•5534P6 ' 


52 


•005205 


•721419 


53 


•002301 


•361965 


53 


•003643 


•561474 j 


53 


•005295 


•723883 


54 


•002321 


•365694 


54 


•003668 


•564442 1 


M 


•00'5325 


•726351 


55 


00-2341 


•369406 


55 


•003093 


•5G7401 i 


55 


•005356 


•7-28812 


56 


00-2361 


•373100 


56 


•003718 


•570349 i 


56 


•0053S6 


•731267 


57 


•002381 


•376779 


57 


•003744 


•573288 


57 


•005416 


•733713 


58 


•002401 


•380445 


58 


•003769 


•576216 i 


58 


•005447 


•736155 


59 


-002422 


-:8t(J93 


59 


•003794 


•579137 1 


59 


•00547S 


•7385^9 ; 


60 


002442 


•387728 


60 


•003320 


•583045 1 


60 


•005508 


•741(rl6 i 

i 
1 



•120 








Yersed 


Sines. 








6 DEGREES. 


7 DEGREES. 


' 8 DEGREES. 


Milt. 


Nat. No. 


j Logarithm, 


Min 


; Nat.. No, 


' LoKHrithm. 


' iMin. 


: Nat No. 


j Logarithm. 


1 


0-005478 


7^738630 





0-007454 


, 7-872380 





0-009782 


7-988199 


1 


•005509 


•741038 


1 


: •0074S9 


; -874444 


1 


•009772 


•990003 


2 


•005539 


•743438 


2 


! ^007525 


, -876502 


2 


•009803 


•991804 


3 


•005570 


•745831 


3 


•007561 


i -873555 


3 


•009854 


•993601 


4 


•005600 


•748220 


4 


: -007596 


! -880603 


4 


•009894 


•995396 


! 5 


•005631 


•750601 


5 


1 -007682 


1 -832647 


5 


•0C9935 


•597185 


6 


j •005662 


•752974 


6 


: -007663 


j -884636 


6 


•009976 


•993972 


7 


•005693 


•755342 


7 


, -007704 


-886719 


7 


•010017 


8-000754 


1 8 


•005724 


•757704 


8 


1 -007740 


-838749 


8 


•010058 


♦002532 


» 


•005755 


•760057 


1 9 


' -007776 


-890773 


9 


1 ^010099 


•004807 


I 10 


•0057S6 


•762406 


I 10 


: -007S13 


-892793 


10 


•010141 


'006079 


11 


•005S13 


•764749 


! 11 


1 -007849 


' -394808 


11 


•010181 


•00T847 


12 


•005849 


•767084 


i 12 


' -007835 


I -S9i)818 


12 


•010223 


•009611 


; 13 


•0C5S80 


•769413 


13 


-007922 


i -898824 


13 


•010265 


•011371 


14 


•005912 


•771738 


14 


-007958 


! -900825 


14 


•010307 


•013128 


: 15 


•005944 


•774055 


15 


•007995 


: -932821 


15 


•010348 


•014S83 


16 


•005975 


•776364 ■ 


16 


-008382 


•934813 


16 


•010390 


•016682 


17 


•006007 


•778671 : 


17 


-008069 


•906800 


17 


•010432 


•018379 


IS 


•006089 


. -780968 1 


18 


-008106 


•908783 : 


18 


•010474 


•020121 


19 


•006071 


•783261 i 


19 


-008143 


-910761 ; 


19 


•010516 


•021861 


20 


•006103 


•785547 i 


20 


-008180 


-912734 


20 


•010558 


•028597 


I 21 


•006185 


•787829 


21 


•008217 


•914704 


21 


•010600 


•025329 


: 22 


•006167 


•7901 02 


22 


-008254 


•916670 


22 


•010643 


•027058 


'28 


•006200 


•792369 


23 


•008291 


•91862S 


23 


•010685 


•028T83 


: 2- 


•006232 


•794633 


24 


-008829 


•920534 ; 


24 


•010728 


•030505 


: 2ia 


•006265 


•796891 i 


25 


-008866 


•922536 ; 


25 


•010770 


•032223 


1 20 


•006297 


•799140 ! 


26 


-008404 


-924483 


26 


•010813 


•033939 


i 27 


•006330 


•801385 : 


27 


•008442 


-926425 : 


27 


•010856 


•035651 


i 23 


•006362 


•803624 


28 


-O0S4T9 


-928863 i 


28 


•010893 


•037359 


: 29 


•006395 


•805859 : 


29 


•008517 


" -980297 ; 


29 


•01C941 


•039064 


! 30 


•006428 


•808086 : 


80 


•008555 


-932227 ' 


30 


•010984 


•040766 


; 31 


•006461 


•810307 1 


81 


•008598 


•934152 : 


31 


•011027 


•042465 


! 82 


•006494 


•812524 i 


32 


•008631 


-936074 


32 


•011070 


•044159 


33 


•006527 


•814734 ' 


33 


•008669 


•937990 


33 


•011113 


•045850 


i 34 


•006560 


•816939 


34 


•008708 


•939903 ; 


34 


•011157 


•047539 


: 35 


•006594 


•819189 1 


35 


•008746 


•911811 


35 


•011200 


•049325 


; 36 


•006027 


•821332 i 


36 


•0C8784 


•943715 • 


36 


•011243 


•050936 


37 


•006661 


•823521 1 


37 


•008828 


•945615 : 


87 


•011287 


•052584 i 


^ 38 


•006694 


•825704 ; 


38 


•008862 


•947511 : 


38 


•011331 


•054*260 


39 


•006728 


•S27S81 ; 


89 


•008930 


•949403 i 


39 


•011374 


•(55931 


40 


•006762 


•880052 ; 


40 


•008939 


•951290 ' 


40 1 


•011418 


•C57601 


41 


•006795 


•832218 i 


41 


•008978 


•953178 ; 


41 I 


•011462 


•059266 


42 


•006829 


•884:379 i 


42 


•009017 


•955(52 i 


42 ' 


•011506 


•06092S 


43 


•006863 


•836585 : 


43 


•OC9056 


•956927 1 


43 


•011550 


•062588 


44 


•006897 


•S386S5 : 


44 


•0C9C95 


•958799 i 


441 


•011594 


•064243 


45 1 


•006932 


•840S30 j: 


45 ; 


•009134 


•960666 


45 1 


•011638 


•065896 


46 


•006966 


•842969 1 


46 


-0C9178 


•962529 1 


46 


•011682 


•067546 i 


47 1 


•007000 


•845115 ■ 


47 


-009213 


-964338 ! 


47 


•011727 


•069192 


4S ! 


•007084 


•847232 ; 


48 


•0C9252 


•96624:3 i 


48 


•011772 


•070S36 


49 


•007069 


•849356 


49 1 


•009292 


•963C94 ! 


49 


•011816 


•072476 


50 


•007104 


•851475 


50 1 


•009331 


•969342 i 


50 


•011860 


•074113 


61 


•007138 


•853589 ; 


51 ! 


•009371 


•971784 i 


51 


•011905 


•075747 


52 


•007173 


•855697 :i 


52 J 


•009411 


-978624 i 


52 


•011950 


•077378 


53 


•007208 


-857800 :: 


53 1 


•0C9450 


-975459 : 


53 


•011995 


•079007 


54 ; 


•007243 


•859393 :' 


54 1 


•009491 


•977291 i 


54 


•012040 


•080631 


55 ! 


•007278 


-861991 il 


55 i 


•0C9581 


-979118 1 


55 


•012385 


•082253 


56 


•007313 


-864079 ] 


56 1 


•0C9552 


•930942 


56 


•012130 


•083872 


57 : 


•007848 


-866162 i 


57 j 


•009311 


-932762 


57 


•012175 


•085488 


58 • 


•007383 


•868240 


58 i 


•0;9j51 


-934578 


58 


•012220 


•087100 1 


59^ 


•007418 


•870312 


59 j 


•0C9GD1 


-93G390 


59 


•012260 


♦08S713 


CO ' 

i 


•007454 


•872880 


60 


•0C9732 


-9SS199 


60 1 


•012311 


•090316 









External 


Secants. 






421 


6 DEGEEES. 


7 DEGEEEH. 


! 8 BEGEEES. 


Min. 


Nat. No 


1 Logarid..,.. 


Mi>. 


Nat. No. 


l.ogaiitliin. 


1 

; Mm. 


Nat No 


LogaritLiii. 





0-005508 


7-741016 





0-007510 


7-875628 





0^0C9828 


7992447 ' 


1 


•005539 


•743437 


1 


-007546 


•877708 


1 1 


•009873 


■994268 i 


2 


•005570 


•745850 


2 


•007581 


-879782 


1 2 


•009910 


■996087 ! 


S 


•005601 


•74S257 


3 


•007618 


-881851 


1 3 


-009952 


•997902 ' 


4 


•005632 


•750659 


4 


•007654 


•883915 


i 4 


•009998 


•999715 ! 


5 


•005663 


•753054 


5 


-007691 


-885974 


5 


•010085 


8-001521 1 


6 


•005694 


•755440 


i 6 


•007727 


•838029 


6 


-010077 


-003826 


7 


'005726 


•757822 


! 7 


•007764 


•890078 


7 


-010119 


•005126 


8 


•005757 


•760197 


i 8 


•007801 


-892124 


8 


•010160 


-006921 


9 


•0057SS 


•762564 


1 9 


•007837 


•894163 


9 


•0102C3 


•008716 


10 


•005820 


•764926 


! 10 


•007874 


•896199 


10 


•C10245 


-01C5C5 


11 


•005852 


•767283 


1 11 


•007911 


•898230 


11 


•010287 


•012292 i 


12 


•005883 


•769632 


i 1'^ 


•007948 


•9'Jl250 


12 


•010329 


•014074 : 


13 


•005915 


•771974 


1 13 


•007985 


-902278 


13 


•010372 


•015852 i 


14 


•005947 


•774813 


i 14 


•0C8022 


•904295 


14 


•01C414 


-017627 ! 


15 


•005979 


•776644 


15 


•008059 


•906807 


! 15 


•010457 


•019401 i 


16 


•006011 


•778967 


16 


•0G8C97 


•908815 


! 16 


•01G499 


•021148 ■ 


17 


•006043 


•781288 


17 


•0(8184 


•910ol8 


1 17 


•01C542 


•022933 


18 


•000076 


•783599 


18 


•008172 


•912318 


^.8 


•01C585 


•C24694 


19 


•006108 


•785906 


19 


•0C821.9 


•914312 


19 


•010628 


•026452 


20 


•006141 


•788206 


20 


•008247 


-916S01 


1 20 


-010671 


•(28207 


21 


•006173 


•790502 


21 


•0C8285 


•918287 


! 21 


•010714 


•029957 


22 


•006206 


•792789 


22 


•0U8323 


-920270 


I 22 


•010757 


•081705 


2a 


•006238 


•795070 


23 


•0C8361 


•922244 


23 


'01G8C0 


•C83449 


24 


•006271 


•797848 


24 


•008899 


•924216 


24 


•01C844 


•035189 


25 


•006304 


•799620 


25 


•0C8437 


•926185 


25 


•OK 887 


•086926 


26 


•006337 


•801S83 


26 


•008475 


•928148 


26 


-OK 981 


•088061 


27 


•006370 


•804143 


27 


•0C8513 


•980107 


27 


•01(975 


-040891 ; 


28 


•006403 


•806396 


28 


•0t8552 


•982061 


28 


•011018 


•C42118 i 


29 


•006436 


•808645 


29 


•018590 


•984012 


i 29 


•011062 


•048842 : 


au 


•006470 


•810887 


80 


•018529 


•935958 


80 


•011106 


•04^563 1 


81 


•006503 


•813122 


31 


•018668 


•937900 


, 31 


•011150 


-C47281 ■ 


32 


•006537 


•815854 


82 


•0C8706 


•939839 


' 32 


•011194 


•C48994 ' 


33 


•006570 


•817578 


33 


•0C8745 


•941771 


33 


•011238 


•(50704 ; 


34 


•006604 


•819798 


34 


•008784 


•948701 


: 84 


•011282 


•052412 


35 


•006638 


•822012 


35 


•0(.-8823 


•945626 


i 85 


•011326 


•C54117 ^ 


86 


•006671 


•824220 


36 


•008862 


-947547 


36 


•011371 


•055817 i 


37 


•006705 


•826423 


37 


•008901 


•949464 


: 37 


•011416 


•057514 1 


38 


•006739 


•828621 


88 


•008941 


-951377 


38 


•011461 


•059209 


39. 


•006773 


•830813 


89 


•008980 


•958286 


39 


•011506 


•060899 


40 


•006808 


•832993 


40 


•009020 


•955190 


40 


•011550 


•062588 


41 


•006842 


•835179 


41 


•009059 


-957090 


41 


•011595 


•CG4273 1 


42 


•006876 


•837355 


42 


•009C99 


•958986 


42 


•011640 


•CC5954 1 


43 


•0069 11 


•839526 


43 


•009189 


-960878 


43 


-011685 


•067683 I 


44 


•006945 


•841691 


44 


•009178 


•962767 


44 


•011730 


•CC98( 8 


45 


•006980 


•843851 


45 


•009218 


•964651 


45 


•011776 


•071 980 1 


46 


•007015 


•846005 


46 


•009258 


-966531 


46 


•011821 


•072650 j 


47 


•007049 


•848155 


47 


•009298 


•968408 


47 


•011866 


-074315 i 


48 


•007084 


•850298 


48 


•009839 


•970280 


48 


•011912 


•075979 ! 


49 


•007119 


•852437 


49 


•009379 


•97214^ 


49 


•011957 


•077688 


50 


•007154 


•854571 


50 


•009419 


•974013 


50 


•012003 


•C79295 


51 


•007189 


•856700 


51 


•009460 


•975873 


51 


-012049 


•C8l 949 


52 


•007225 


•858323 


52 


•009500 


•977730 


52 


-012U95 


•082599 


53 


•007260 


•860942 


53 


•009541 


•979583 


53 


•012140 


•084248 


54 


•007295 


•863055 


54 


•009581 


•981432 


54 


-012187 


•085892 


55 


•007331 


•8')51()3 


55 


•009622 


•98:]277 


55 


-012232 


•087534 


56 


•007367 


•867267 


56 


•009663 


•985119 


56 


-012279 


•189172 


57 


•007402 


•869365 


57 


•009704 


•9S6956 


57 


•012825 


•C9(8i:8 


58 


•007438 


•ST1458 


58 


•009745 


•988790 


58 


•012372 


•( 92440 


59 


•007474 


•873546 


59 


•0C9786 


■990619 


59 


•012418 


•(940(50 1 


60 


•007510 


•875628 


"' 


•009828 


•992446 


60 


•012465 


•(95696 



30 



422 



Versed Sines. 



9 DEGREES. 1 


10 DEGEEES. | 


1 
11 DEGREES. ! 


Min. 1 Nat No 


LogHiifhm '! M 


in. i 


Nat. No. 1 


Logarithm. | 


.Aiin. 


Nat No 


L.)g;ui.hm. 





0-012311 


8-090316 i 





0-015192 


8-181622 





0-C18378 


8-264176 : 


1 


•0121357 


•0919-20 i 


1 


•015242 


•183065 


1 


•01342S 


•265486 i 


2 


•012403 


•C93521 ; 


2 


-015293 


•184505 


2 


•018484 


•266796 1 


8 


•C1244S 


•095119 : 


8 


•015349 


•185943 


8 


•018541 


•268183 1 


4 


•012494 


•C9.>714 i 


4 


•015395 


•187378 


4 


•018596 


•2e94C.7 ! 


5 


•012540 


•098306 • 


5 


•015446 


•188811 


5 


•018651 


•27*1711 i 


6 


•012580 


•099894 


6 


•015497 


•190242 


6 


•0187(»7 


•272012 1 


7 


•012632 


•10148) ■ 


7 


•015548 


•191671 


7 


•018762 


•273293 ; 


8 


•012378 


•103064 1 


8 


•015599 


-193097 


I 8 


•018819 


•2746:8 ! 


9 


•012724 


•104644 


9 


•015650 


•194520 


9 


•018376 


•27c9 4 


10 


•012770 


•106-221 1 


LO 


•015701 


-195942 


1 10 


•018932 


•277197 i 


11 : 


•012817 


•107793 1 ] 


LI 


•015752 


•197361 


! 11 


•018988 


•278487 i 


12 


•0128(54 


-109367 ; 1 


2 


•G15804 


•198778 


i 12 


•019045 


•279777 


13 


•012910 


-11C936 1 


[3 


•015856 


•200192 


1 13 


•019101 


•231065 


14 


•012957 


■112531 1 


L4 


•0159:;8 


•201604 


1 14 


•019158 


■232350 


15 


•013003 


•114065 ':[ : 


L5 


•015959 


•203014 


: 15 


•019215 


•2S8684 


16 


•013050 


•115625 


L6 


•016011 


•204421 


1 1^ 


•019272 


•284915 


17 


•013097 


•117182 i| - 


L7 


•016363 


•2C5826 


17 


•019828 


•286194 


18 


•013144 


•118737 h ■ 


L8 


•016115 


•207229 


i 18 


•019385 


■287474 


19 


•013191 


•12;i238 ;l " 


19 


•016167 


•2C8639 


i 19 


•019442 


•238749 


20 


•013238 


•1218^8 : 


10 


•016219 


•210028 


20 


•019499 


•290023 


21 


•013286 


•123384 ! 


21 


•016271 


•211424 


i 21 


•019557 


•291296 


22 


•013333 


•124927 i 


22 


•016323 


■212327 


22 


•019614 


-292566 


23 


•013383 


•126468 j 


28 


•016376 


-2142.9 


28 


•G19671 


•2938:35 


24 


•013423 


•128306 1 


24 


•016428 


•215598 


24 


•019729 


-295101 


25 


•013475 


-129542 


25 


•016481 


-216986 


25 


•019736 


•296366 


26 


•013523 


-131074 


26 


^016533 


•218:371 


26 


•019344 


•2976-29 


27 


•013570 


-132303 1 


27 


•016586 


•219758 


27 


•019902 


•293839 


23 


•013'318 


-134131 \ 


28 


•016639 


•221134 


: 28 


•019959 


•890149 


29 


•013366 


-135635 1 


29 


•016692 


•222502 


29 


•020C17 


•3004C6 


30 


•013T14 


•137176 ■ 


30 


•016745 


-223887 


SO 


•020075 


•302661 


31 


•013762 


-138695 : 


31 


•016798 


-225261 


81 


•02C133 


•303916 


32 


•013810 


•140212 H 


33 


•016851 


-2266:38 


82 


•020191 


•S051€7 


33 


•013859 


-141726 j 


33 


•0169 J4 


•228002 


88 


•02C250 


•806417 


84 


•013907 


-14:3238 \\ 


34 


•01 0958 


•229370 


1; 84 


•020308 


•807666 


35 


•013955 


•144745 li 


35 


•017011 


-230734 


!: 85 


•020366 


•308912 


86 


•014003 


i -146251 ' 


36 


•017065 


•232397 


36 


•023425 


•810156 


37 


•014052 


1 -14:754 


37 


•017118 


•233458 


37 


•020483 


•811899 


38 


•014101 


•149255 


38 


•017171 


-234817 


38 


•023541 


•812680 


39 


•014149 


' -153752 '; 


39 


•017225 


-236178 


89 


•02C600 


•31888) 


40 


•014193 


! -152248 1 


40 


. ^017279 


•237528 


: 40 


•G20659 


•815117 


41 


•014247 


1 -153741 :; 


41 


•017338 


-288880 


41 


•020718 


•816352 


42 


•01429G 


i -155231 i 


42 


•017387 


•240230 


i 42 


•C20777 


•817537 


43 


•014:345 


i -156719 ; 


48 


•017441 


•241578 


: 43 


•02G836 


•318-318 


44 


•C14394 


1 -158238 ! 


44 


•Cn[7496 


•242924 


:, 44 


1 •023895 


•8200^ 


45 


•01444:3 


1 -159386 . 


45 


•017550 


•244267 


1 45 


•023954 


•821278 


46 


•014493 


i -161165 ; 


46 


•017604 


•2456C9 


46 


•021014 


•822535 


47 


•014542 


1 -162343 i 


47 


•017658 


•246949 


i 47 


•021078 


•8237:30 


4S 


•014592 


•164118 ! 


48 


•017712^ 


•248286 


li 48 


•G21183 


•824953 


49 


•014641 


! -165589 j 


49 


•017767 


•249621 


1' 49 


•G21192 


•326174 


5. 


•014691 


, -16706 ) i 


50 


•017822 


•25G955 


1 50 


•021-252 
•021311 


•827S95 


51 


•014741 


1 -163527 \ 


51 


•017377 


•252-236 


1^ 51 


•328618 


i 52 


•014791 


-169992 : 


52 


•017981 


•253615 


|. 52' 


•021371 


•329829 


i 58 


•014841 


; -171454 , 


53 


•017986 


•254942 


!■ 58 


•021431 


•831044 


i 54 


•014S91 


■172914 ! 


54 


•018341 


•256267 


54 


•021491 


•332256 


\ 55 


•014941 


•174372 ; 


55 


•018096 


•257591 


1 55 


•021551 


•8S:34G9 


' 56 


•014991 


; -175327 • 


56 


•013151 


•253911 


i 56 


•021 on 


•334678 


57 


•015041 


! -177-279 i! 


57 


•318206 


•260230 


57 


•021671 


•835885 


: 53 


'O15091 


! -1787:^9 :; 


53 


•018262 


•261548 


58 


•021782 


•337f.92 


59 


•015141 


1 -18 "^117 i 


59 


1 •018317 


•262362 


59 


•021792 


•338296 


i 60 

i 


•015192 


-1816-2;^ i 


60 


' •018378 


•264176 


li "" 


•C21852 


1 -3:39499 



External Secants. 



423 



9 DEGREES. 




10 DEGEE^S. 1 


11 DEGEEES. 


Mill. 


N.t. No. 


Loganthin. 


Mm. 


Nat No. j 


L,.s.nthm. 1 


Mill. 


Nat. No. 


1 

Losarilhm j 





0-012465 


8-095696 i 





©•G15426 


8188271 





0^018717 


8-272229 


1 


•012512 


•C97820 


1 


•015478 


•189732 


1 


•018774 


•273564 


2 


•012.559 


•093941 


2 


•015530 


•191193 


2 


•018832 


•274899 ' 


8 


•01261)5 


•100559 


3 


•015583 


■ ^192658 


8 


•018S91 


•276260 


4 


•012652 


•102174 1 


4 


•015686 


•194116 


4 


•01S948 


•277559 


5 


•012699 


•103787 1 


5 


•015688 


•195571 


5 


•019006 


•278888 


6 


•012746 


•105395 1 


6 


•015740 


•197025 


6 


•019064 


•280213 


7 


•C12794 


•107001 


7 


•C15793 


•193476 


7 


•019122 


•281527 


8 


•012S41 


•1086iJ5 I 


8 


•015846 


•199925 


8 


•019180 


•282859 





•012889 


•1102iJ6 i 


9 


•015S99 


•201371 


9 


•019239 


•284180 


10 


•012936 


•111804 1 


10 


•015952 


•202815 


10 


•019297 


•285494 


11 


•012984 


•113399 ! 


11 


•016005 


•204257 


11 


•019-356 


•286818 


12 


•018081 


•114990 1 


12 


•016058 


•205697 


12 


•019415 


•288128 


13 


•013079 


•116579 ! 


13 


•016111 


•207138 


18 


•019473 


•289441 


14 


•013127 


•118165 i 


14 


•016164 1 


•208568 


14 


•0195.32 


•290751 


15 


•018175 


•119749 1 


15 


•016218 


•210001 


15 


•019591 


•292060 


16 


•013223 


•121330 1 


16 


•016271 


•211431 


16 


•019650 


•293366 


IT 


•013271 


•1229^:8 ! 


17 


•016325 


•212859 • 


- 17 


•019709 


•294670 


18 


•01.3319 


•124483 i 


18 


•016379 


•214285 


18 


•019769 


•295976 


19 


•013367 


•126055 


19 


•016433 


•215718 


19 


•019828 


-297276 


20 


•013416 


•127626 


20 


•016486 


•217130 


20 


•019837 


•298575 


21 


•013464 


•129193 


21 


•016540 


•218549 


21 


•019947 


•299874 


22 


•01.3513 


•130756 i 


22 


•016594 


•219965 


22 


•020006 


•301169 


28 


•013561 


•132318 ' 


23 


•016649 


•221380 


23 


•020066 


•3C2463 


24 


•013610 


•1.33877 i 


24 


•016703 


•222T92 


24 


•020126 


•308755 


25 


•013659 


•135434 i 


25 


•016757 


•224203 


25 


•C20186 


-305045 


26 


•013708 


•136937 1 


26 


•016811 


•225612 


26 


•020246 


-806334 


27 


•013757 


'138537 i 


27 


•016866 


•227017 


27 


•020306 


•307619 


28 


•0138.)6 


■'140086 ! 


28 


•016920 


•2-28421 


28 


•C2C366 


-808905 


29 


•013856 


•141631 


29 


•016975 


•229922 


29 


•020426 


•310183 


80 


•0139D5 


•143174 : 


80 


•017030 


•231-221 


80 


•020486 


•311469 


31 


•013954 


■144714 


31 


•017085 


•232618 


81 


■•(;20547 


•312749 


32 


•014004 


■UG2b2 • 


82 


•017140 


•234014 


82 


•020608 


•314026 


38 


•014054 


•147787 


38 


•017195 


•235406 


88 


•020668 


•815302 


34 


•014103 


•140318 


34 


•017250 


•236798 


34 


•020-729 


•316576 


85 


•014153 


•150819 


85 


•0173C5 


•2.38185 


35 


•020790 


•317848 


86 


•0142v;8 


•152376 I 


86 


•017860 


•239572 


36 


•02:851 


•319118 


37 


•014253 


•153900 ; 


87 


•017416 


•2-!' 957 


87 


•02.912 


•320887 


88 


•014332 


•155423 : 


88 


•017472 


•242339 


83 


•020973 


•32164^4 


89 


•G14353 


•150941 


89 


•017527 


■243719 


89 


•021034 


•322920 


40 


•014403 


•158458 


40 


•017582 


•245:98 


40 


•021C95 


•324183 


41 


•014458 


•159973 


41 


•017638 


•240474 


41 


•021156 


•825444 


42 


•014508 


•161485 


42 


•C17GD5 


-247848 


42 


•021218 


•326705 


43 


•014554 


•162094 


43 


•017751 


"249219 


43 


•021-280 


•327963 


44 


.•0146a5 


•164500 


44 


•0178.7 


•250580 


44 


•021341 


•829-220 


45 


•014655 


•166004 


45 


•017803 


•251956 


45 


•021408 


■330475 


46 


•014706 


•167505 


46 


•017910 


•253322 


46 


•021465 


■331728 


47 


•014757 


•169005 


47 


•017975 


•254686 


47 


•021527 


•332979 


48 


•0148)8 


•170502 


48 


•018.32 


•256047 


48 


•021539 


•3:34229 


49 


•014859 


•171905 


49 


•01S3S0 


•257407 


49 


•021051 


•335477 


50 


•0149.19 


•1734SS 


50 


•018145 


•258765 


50 


•021713 


•386724 


51 


•C14961 


•174977 


! 51 


•01S202 


•260120 


51 


•021776 


•8:37969 


52 


•015018 


•173464 


1 52 


•018-258 


•261473 


52 


•021838 


•839211 


58 


•015064 


•17794S 


53 


•01 S3 15 


•262825 


53 


•021900 


•340453 


54 


•C15115 


•179430 


54 


•018372 


•264174 


54 


•021968 


•841091 


55 


•015167 


•18:910 


55 


•018'i;30 


•265522 


55 


•022026 


•342981 


56 


•015218 


•1S238T 


56 


•0-18487 


•266867 


56 


•022089 


•:344167 


57 


•015270 


•183S31 


57 


•018544 


•268210 


57 


•022151 


•84540i> 


58 


•015322 


1 •18534:3 


58 


•018601 


•269552 


58 


•02-2214 


•846084 


59 


-015874 


•18:38 -S 


59 


•018659 


•270891 


59 


•022277 


•847865 


60 


•015426 


•18S271 


60 


•018717 


•272220 


00 


•028341 


•349005 



424 








Versed 


Sines. 








12 DEGEEES. 


1 


3'DEGREES. | 


14 DEGEEES. 


Mill. I 


Naf, Nn. i 


Lopa-ilm-. i 


Vtn. 1 


Nat No 1 


I.o!;«rilbm. i 


Min. 1 


Nat. No. 


L..garifhin 


' 


0-021852 


8-S£9499 


1 


0^C25680 


8408747 


: 


0-C29704 1 


8-472819 


1 i 


•G21913 


•840700 1 


1 


•025695 


•4(9556 


1 ' 


•C29775 1 


•47384S 


2 


•C21974 


•341900 ' 


2 


•(25761 


•41(962 


2 


•C29S45 ! 


•474874 


3 


•G22034 


•343C97 i 


3 


•C25S27 


•412tG7 


3 


•C29916 ' 


•475699 


4 


•C22C95 


•344293 : 


4 


•C25S92 


•413171 ' 


4 


•G-299S6 ' 


•476925 


5 


•C22156 


•845488 1! 


5 


•025958 


•414273 ■ 


5 


•030057 


•477948 


6 


•022217 


•346681 ; 


6 


•C26024 


•415;374 


6 


•G80127 


•478970 


7 


•02227S 


•847S77 


7 


•C26(90 


•41 6474 


7 


•080199 


•479991 


8 


•02233S 


•349062 


8 


•C26156 


•417573 


8 


•C3C270 


•481C11 


9 


•02240C ! 


•350249 


9 


•C26222 


•41S6G9 


9 


•030341 


•482029 


10 


•C22461 1 


•351435 


10 


•C262S8 


•419764 


10 


•08C412 


•483046 


11 


V22523 I 


•352620 


11 


•C26S55 


•420858 


11 


•030483 


•484062 


12 


•022584 ; 


•:3558C2 : 


12 


•C26421 


•421951 


12 


•C8C555 


•485078 


13 


•022646 


-3549S4 ! 


13 


■•:;26498 


•428042 


13 


•08r:626 


•486091 


14 ! 


•022707 


•35G163 ; 


14 


•026564 


•424181 


14 


•080 C97 


•487108 


15 ! 


•022769 


•857342 


15 


•026621 


•425219 


15 


•08 7G9 


•483115 


16 1 


•C22831 


-85S51S ^ 


16 


•(23687 


•4268' 9 


16 


•(8.841 


-4S9125 


17 


•022S92 1 


•359G93 : 


17 


•G26754 


•427898 


17 


•C8 912 


-490183 


18 


•022954 1 


•360867 ; 


18 


•C26821 


•428477 


18 


•C8r9S4 


•491141 


19 ' 


-023016 j 


•362089 ' 


19 


•G26SS3 


•429.!^)0 


19 


•031(56 


•492143 


20 i 


•0230T9 


•3G32G8 : 


20 


•C26955 


•430641 


20 


•031128 


-49815^3 


21 


•023141 i 


•364376 ; 


21 


•027022 


•4^17^22 ! 


21 


•081200 


•494157 


22 1 


•023203 


•865543 


22 


•C27G89 


•482800 ! 


22 


•G81272 


•49516G 


23 


•023266 


•366719 


28 


•C27157 


•433877 


23 


•GS1S45 


•496162 


24 , 


•023323 


•867S72 


24 


•i 27224 


•484954 


24 


•031417 


•49n62 


25 


•023390 


•869085 


25 


•C27292 


•48GC29 


25 


•081489 


•498162 


26 


•023453 


•87ai95 


26 


•C27359 


•487102 


26 


•C31562 


•499160 


2T 


•0-23515 


•8718^ ■ 


27 


•027427 


•488174 


27 


•C81634 


•500158 


28 


•023578 


•872511 


28 


-027494 


•489244 


28 


•031707 


•501153 


29 


•023641 


•873667 


29 


•027562 


•440814 


29 


•081780 


•502148 


30 


•0-23704 


•374822 ; 


30 


-027630 


•441382 


30 


•081852 


•5C3142 


31 


•02S7G7 


•875974 : 


31 


-027693 


•442449 


31 


•081925 


•504134 


32 


•023530 


•827125 • 


32 


-C27766 


•44:3514 


32 


•C81998 


•505125 


33 


•023S93 


•878275 


33 


-C27834 


•444578 


33 


•082071 


•506115 


34 


•023956 


•879423 : 


84 


•C27992 


•445(>41 


34 


•032144 


•507105 


35 


•024020 


•88C569 


35 


-027971 


•446702 


35 


•032218 


•5C8C93 


36 


■024083 


•381715 


36 


•(^23089 


•447762 


36 


•082291 


•5C9079 


37 


-0-24147 


•882858 


87 


•G28107 


•44SS21 


37 


•082364 


-510C65 


38 


•024210 


•3S40il 


38 


•028176 


•449878 


38 


•C8'2488 


•511C49 


89 


•G24279 


•835141 


89 


•028245 


•45 985 


89 


•032511 


•012032 


40 


•024338 


•8SG279 


40 


•02S313 


•451990 


40 


•C325S5 


•513014 


41 


•024402 


•887417 


41 


•028882 


•453043 


41 


•032659 


•518996 


42 


•024465 


•888553 


42 


•0-28451 


•454096 


42 


•032732 


•514976 


43 


•024529 


•3S36S7 


43 


•G28520 


•455147 


43 


•032806 


•515^5 


44 


-024594 


•890821 


44 


•C285S9 


•456196 


44 


•082S80 


•516933 


45 


•024558 


•391952 


45 


•028658 


•457244 


45 


•082954 


•517909 


46 


•024722 


•893082 


46 


•023727 


•458291 


46 


' -08:3028 


•5188^ 


47 


•0247S6 


! .894210 


47 


•028796 


•459383 


47 


•033102 


; -519858 


48 


•024851 


! •895833 


48 


•023866 


•460882 


48 


: •038177 


1 -520832 


49 


•024915 


; •39(>463 


49 


•028985 


•461425 


49 


•033251 


i •521804 


50 


•024930 


1 •397587 


50 


•G290C5 


•46*2463 


50 


! -033325 


; •522n5 


51 


•025044 


i •893710 


51 


•G29074 


-463503 


51 


i -033400 


' •523745 


52 


•C251.9 


•899831 


52 


•029144 


-464547 


52 


' -033475 


i -524714 


53 


•025174 


1 •400951 


58 


•C29214 


•465586 


53 


-033549 


•525681 


• 54 


•C25439 


•402069 


54 


•029283 


•466G28 


54 


-038624 


•526648 


55 


•025304 


•4C3185 


55 


•029358 


•467659 


55 


: -038699 


' •527614 


56 


•025369 


•404S0O 


1 56 


•02942:3 


•468693 


56 


•033774 


'■ ^52^78 


67 


•025434 


•4C5414 


i 57 


•029493 


-469726 


57 


' -033849 


i ^529541 


58 


•025499 


•406527 


' 58 


•G29564 


•470759 


58 


•033924 


' 530504 


59 


•025564 


i •407637 


1 59 


•C29634 


•471789 


59 


•033999 


I •581465 


i 60 

1 


•025630 


; -408747 

i 


1 60 


•029704 


-472819 


, 60 


•034074 


1 -532425 



External Secants. 



425 



Min 


12 DEGREES. 


I 
! 


13 DEGEEES. 


14 DEGREES. 


N;it. N.i, 


I ..„..i.,.n. 


; V.-.V. 


i Nat. No. 


i i,o..aM Inn. 


Mill. 


i Nat. No. 


Logarithm. 





0-02-234t 


i 8-349 J95 


\ ^ 


! 0^0233l)4 


8-420U23 





0^030614 


8-485915 


1 


•022434 


i -350322 


1 


1 -023373 


•421161 


1 


-030688 


•483975 


2 


•022467 


' •351540 


! 2 


-026442 


•422293 


2 


•030763 


•488033 


3 


•022531 


•352773 


i 3 


-026511 


•4234:31 1 


3 


•030838 


•4S9C90 


4 


•022594 


•353996 


i .4 


•026581 


•424564 1 


4 


•03G913 


•490147 


5 


•022658 


•355218 


1 5 


•026650 


•425395 ! 


5 


•030938 


•491202 


6 


•022722 


•356433 


6 


•026720 


•426323 1 


6 


-031064 


•492256 


T 


•022786 


•357662 


'^ 


•026789 


•427955 : 


T 


•031139 


•493308 


8 


•022349 


•358374 


! 8 


•023359 


-429084 ; 


8 


•031215 


•494360 


9 


•022913 


•360088 


1 9 


•026928 


-430209 ; 


9 


•031290 


•495410 


10 


•022977 


•861301 


i 10 


•023998 


•431334 , 


10 


•031366 


•493459 


11 


•023042 


•362513 


1 11 


•02T068 


•432457 : 


11 


•031442 


•497507 


12 


•023106 


•363723 


i 12 


•027138 


•433580 ; 


12 


•031518 


•493555 


13 


•023170 


•364932 


' 13 


•027208 


■434700 : 


13 


-031594 


•499600 


U 


■023235 


•336133 


! 1-t 


•027278 


•435819 : 


14 


-031G70 


•500644 


15 


•023299 


•367345 


15 


•C27349 


•438937 i 


15 


•G31743 


•50168?> 


16 


•023364 


•36S54S 


; 16 


•027419 


•438055 j 


16 


•031822 


•502733 ■ 


17 


•023429 


•369751 


' 17 


•027493 


•439170 ! 


17 


•031899 


•503770 


18 


•023994 


•370952 


18 


•027560 


•440284 ■ 


18 


•031975 


•504810 


19 


•023559 


•372152 


19 


•027331 


•441397 : 


19 


•032052 


•505849 


2) 


•023324 


•373348 


2J 


•027702 


•442508 


20 


•032128 


•50C837 ' 


21 


•023689 


•374544 , 


21 


•027773 


•443319 


21 


•032205 


•5C7923 i 


22 


•023754 


•375739 ' 


22 


•02 7844 


•4447-27 


22 


H)32282 


•508953 ' 


23 


•023820 


•376942 ! 


23 


•027915 


•44-5834 ^ 


23 


•032359 


•5:9993 ; 


24 


•023885 


•378123 i 


24 


•027986 


•446941 . 


24 


•032436 


•511025 , 


25 


•023950 


•379314 1 


25 


•028957 


•443046 : 


25 


•03-2513 


•512058 


2(5 


•024016 


'380502 


26 


•028129 


•449149 = 


26 


•032590 


•513083 i 


27 


•02-i'J82 


•331639 


27 


•028200 


•450252 1 


27 


•032068 


•514119 ! 


23 


•024148 


•332374 


23 


•028272 


•451352 ' 


23 


•032745 


•515146 i 


29 


•024214 


•384058 j 


29 


•023343 


•452452 [ 


29 


•032823 


•516174 1 


30 


•024280 


•385240 1 


3) 


•02S415 


•453551 ; 


30 


•032900 


•517200 1 


31 


•021346 


•380421 : 


31 


•02^^437 


•454648 


31 


•032978 


•518225 ! 


32 


•024412 


•387600 ^ 


32 


•02S559 


•455743 : 


32 


•033056 


•519249 i 


33 


•024478 


•338778 ; 


33 


•028331 


•456838 ' 


33 


•033134 


•520272 ! 


34 


•024544 


•389954 1 


34 


•028703 


•457931 


34 


•033212 


•521294 i 


35 


•024311 


•391128 1 


35 


•023775 


•459;>25 


35 


-033290 


•522315 1 


3o 


'024378 


•3923)2 1 


33 


•0232 IS 


•460113 


33 


-033368 


•523334 


37 


•024744 


•393474 i 


87 


•028920 


•461203 ; 


37 


•033447 


•524353 


3S 


•024811 


•394345 1 


33 


•023993 


•462290 ' 


38 


•033525 


•525370 


39 


•024S78 


•395813 ; 


39 


•029335 


•463378 


39 


•033604 


•526886 


40 


•024945 


•393979 ; 


40 


•029133 


•464464 


40 


•033682 


•527401 


41 


•025012 


•398146 1 


41 


•029311 


•435547 : 


41 


-033761 


•528416 


42 


•025379 


•399310 


42 


•,:.29284 


•4(')6631 


42 


-033840 


•529429 1 


4^3 


•025146 


•400473 


43 


•029357 


•467713 


4:3 


-033919 


•530441 i 


44 


•025214 


•401635 i 


44 


•029430 


•468793 ' 


44 


•033998 


•531453 1 


45 


•025281 


•4:;2795 


45 


•029503 


•469S72 ' 


45 


•034077 


•532463 


46 


•025348 


•403954 


43 


•029577 


•470950 


46 


•034156 


•533470 


47 


•025416 


•405110 


47 


•029650 


•472023 ■ 


47 


•034236 


•534478 


4S 


•025484 


•406237 ; 


48 


•029724 


•478103 : 


48 


-034315 


•535485 ! 


49 


•025552 


•407421 j 


49 


•029797 


•474177 ' 


49 


•034395 


•53{>490 


50 


•025620 


•408573 ! 


5') 


•029871 


•475251 


50 


•034474 


•537495 


51 


•025683 


•409725 1 


51 


•029945 


•476322 


51 


•034554 


•538498 


52 


•025756 


•410S75 1 


52 


•030019 


•477392 j 


52 


•0^34634 


•539501 


53 


•025S24 


•412024 1 


53 


•033093 


•478462 ' 


53 


•034714 


•540501 


54 


•025892 


•413171 


54 


•030167 


•479031 ! 


54 


•034794 


•541502 


55 


•025961 


•414316 


55 


•033241 


•480593 ! 


55 


•034874 


•542501 


56 


•026029 


•415400 


56 


•030315 


•481663 i 


66 


•034954 


•54U99 


57 


•026093 


•416603 


57 


•030390 


•482728 i 


67 


•035035 


•544496 


5S 


•026166 


•417745 


53 


•030464 


•483792 


6S 


•0.35115 


•545493 


59 


•026235 


•418334 


59 


•030539 


'484853 


59 


•085195 


•546437 


CO 


•026304 


•420023 


60 


•030614 


•435915 


60 


•086276 


•547481 

i 



426 








Versed 


Sines, 










15DEGEEES. 


j Min. 


16 DEGREES. 


17 DEGREES. 


Min. 

1 


! Nat. No. 


Logarillim. 


1 Nat. No. 


j Lo-jarithni. 


i Alin. 


I Nat No. 


j Logarilhm. 


1 


i 0-034074 


9-532425 


1 o^ossrss 


8-588140 


; 


j 0-043695 


8-640434 


1 


! -034150 


•533384 


1 


•C8S818 


•599u3S 


1 


-04378J 


-641279 


2 


' -034225 


•584842 


2 


1 •038899 


•599936 


2 


•043865 


642123 


8 


•034300 


•535299 


3 


j ^088979 


! -590888 


3 


•048951 


-642966 


4 


' -0:^376 


-536255 


4 


-039060 


i -591728 


4 


•044036 


•643809 


5 


•034452 


-537210 


5 


! •G89140 


' ^592623 


5 


•044121 


•644650 


i 6 


' -034527 


•588163 


6 


i ^089221 


1 -593517 


6 


•044207 


•645491 


1 T 


•034603 


•539116 


7 


1 •039801 


! -594409 


7 


•044292 


•646880 


8 


; -034679 


-540068 


8 


•089382 


•595301 


8 


•044878 


•647169 


9 


•034755 


-541018 


9 


•089463 


•596193 


9 


•044464 


•648008 


: 10 


•034831 


-541968 


10 


•039544 


•597083 


10 


•044550 


•648845 


11 


•034907 


•542916 


11 


•039625 


•597971 


11 


•044636 


•649682 


12 


•034983 


•543863 


12 


•039706 


•598859 


12 


•044722 


•650517 


13 


'035060 


•5448C9 


13 


•039787 


•599747 


13 


•044808 


•651852 


14 


•035136 


-545755 1 


14 


•039369 


•600633 


14 


•044894 


•652187 


15 


•035213 


-546699 ^ 


15 


•089950 


•601518 


15 


•044980 


•653020 


13 


•035289 


•547642 1 


16 


•040032 


•602403 


16 


•045066 


•653852 


17 


•035!366 


-548584 


17 


•049118 


•603286 


17 


•045158 


•654683 


18 


•035443 


•549525 i 


18 


•040195 


•604169 


18 


•045239 


•655514 


19 


•035520 


-550486 1 


19 


•040276 


•605:51 


19 


•045326 


•656845 


: 20 


•035596 


•551405 1 


23 


•040858 


•605931 


20 


•045412 


•657174 


21 


•035073 


-552342 


21 


•040440 


•606811 I 


21 


•045499 


•658002 


; 22 


•035750 


-553279 ■ 


22 


•040522 


•607090 i 


22 


•045586 


•658880 


. 23 


•035827 


•554215 


28 


•040604 


•608568 


23 


•045673 


•659657 


24 


•035905 


•555150 


24 


•040^86 


•609445 : 


24 


•045760 


•660485 


25 


•035982 


-556034 


25 


•040768 


•610322 


25 


•045847 


•661808 


26 


•036059 


•557017 ; 


26 


•040850 


•611196 i 


26 


•045934 


•662182 


2T 


•036137 


•557948 : 


27 


•04G938 


-612071 1 


27 


•046021 


•662956 


28 


-036214 


•553880 : 


28 


•C41015 


■612945 


28 


•046109 


•663779 


29 


-036293 


•559309 


29 


•041C93 


-613817 ! 


29 


•046196 


•664601 


30 


•036370 


•560738 ^ 


80 


•041183 


•614689 1 


30 


•046288 


•665422 


31 


•036447 


•561665 i 


31 


•041233 


-615560 ' 


81 


•046370 


•666242 


32 


•036525 


•562592 i 


32 


•041346 


•616430 


32 


•046458 


•667061 


83 


•036603 


-5G3518 i 


33 


•041428 


•61729D 


33 


•046546 


•667881 


34 


•036681 


•564448 i 


34 


•041511 


■QISIQT ' 


34 


•046633 


•668698 


85 


•086759 


-565366 ' 


35 


•041594 


-619035 i 


35 


•046721 


•669516 


36 


•036337 


-566289 ! 


86 


-041678 


•619901 i 


36 


•046809 


•670833 


3T 


•036916 


•56T211 ' 


37 


-041761 


•620766 : 


37 


•046897 


•671148 


sa. 


•036994 


•568182 i 


88 


-041844 


•621681 


88 


•046985 


•671963 


39 


•037072 


•569052 i 


39 


•041927 


•622495 ' 


89 


•047078 


•672776 


40 


•087151 


-5G9971 1 


40 


•042010 


•628357 ! 


40 


•047162 


•673590 


41 


•037230 


•570888 1 


41 


•042094 


•624220 ! 


41 


•047250 


•674403 


42 


•037308 


•571805 1 


42 


•042178 


•625081 1 


42 


•047839 


•675215 


48 


•037387 


•572721 t 


43 


•042261 


•625941 1 


48 


•047427 


•676025 


44 


•037466 


•573635 i 


44 


•042345 


•626800 1 


44 


•047516 


•676886 


45 


•087545 


•574550 ! 


45 


•042429 


•627659 


45 


•047604 


•677646 


46 


•037624 


•575462 ! 


46 


•04-2513 


•628517 


46 


•047693 


•678454 


47 


•037708 


•576374 ; 


47 


•042597 


•629378 


47 


•047782 


•679262 


48 


•037782 


•577285 ! 


48 


•042681 


•630230 1 


48 


•047871 


•680069 


49 


•037861 


•578194 ! 


49 


•042765 


•631085 i 


49 


•047960 


•680875 


50 


•037940 


•579103 ; 


50 


•042349 


•631939 1 


50 


•043049 


•681681 


51 


•038020 


•580012 : 


51 


•042933 


•632792 


51 


•048138 


•682485 


52 , 


•038100 


•580919 ; 


52 


•048017 


•633645 


52 


•048227 


•683290 


58 


•038179 


. ^581825 1 


58 


•043102 


•634496 ' 


58 


•048316 


•684098 


54 


•038259 


•582780 \ 


54 


•043186 


•635347 i 


54 


•048406 


•684896 


55 


•038838 


•583634 i 


55 


•043271 


•636197 i 


55 


•04S495 


•685698 


56 


•038418 


•584537 i 


56 


-043355 


•637046 i 


56 


•04S584 


•686498 


57 


•038498 


•585439 1 


57 


-043441 


•t537S95 1 


57 


•048674 


•687298 


58 


•03S578 


•586841 j 


58 


•043525 


•688742 1 


53 


•048764 


•688098 


59 


•038658 


•587241 


59 


•043610 


•639588 i 


59 


•048854 


•688896 


60 


•038738 


•588140 


60 


•043695 


•640434 j 


60 


•048944 


•6S9694 



External Secants. 



427 





15 DEGREES. 




16 DEGREES. 


! 


17 DEGREES. 


Min. 


Nat. No. 


Logarithm. 


! Min. 


1 Nat. No. 


I.»Ka,ilhn.. 


i Mm. 


Nat No. 


Logarithm. 





0-035276 


8-547481 





0-040299 


3-605299 


! 


0-045692 


8^659838 


1 


•035357 


•548473 


1 


•040386 


•606233 


1 


-C45785 


•660721 


2 


•035438 


-549466 


2 


-040473 


•607167 


2 


•045878 


•661604 


3 


•035519 


-550457 


3 


•040660 


•608100 


3 


-C45971 


•662486 


4 


•035600 


•551447 


4 


-040647 


•619332 


4 


•C46C65 


•663867 


5 


•035681 


•552436 


5 


-040735' 


•6C9963 


5 


•C46158 


•664247 


6 


•035762 


-553423 


6 


-040822 


•61C893 


6 


-C46252 


•665127 


7 


•03584-3 


-554400 


7 


-04C909 


-611822 


7 


-046345 


•666CC5 


8 


•035925 


-555396 


8 


-040997 


•612750 


8 


•046439 


•666F83 


9 


•036006 


•556381 


9 


•041085 


•613679 


9 


•C46533 


•667761 


10 


•036088 


•557364 


10 


-041172 


•614605 


10 


-046627 


•668687 


11 


•036170 


•558347 


1 11 


-041260 


•615530 


11 


-046721 


•669513 


12 


•036252 


•559328 


i 12 


-041348 


•616455 


12 


-046815 


-670887 


13 


•036334 


•560309 


1 13 


•041437 


•617380 


13 


-04C910 


•671261 


14 


•036416 


•561289 


i 14 


•041524 


•6183G2 


14 


•C47004 


•672185 


15 


•036498 


•562267 


: 15 


•041613 


•619224 


I 15 


-047099 


•678008 


16 


•036580 


•563245 


i 16 


•041701 


•620146 


1^16 


-047193 


•673879 


IT 


•036662 


•564221 


1 17 


•041789 


•621066 


1 1^ 


-047288 


•674749 


18 


•036745 


•565197 


1 18 


•041878 


•621982 


I 18 


-047883 


-675619 


19 


•036828 


•566172 


i ^^ 


•041967 


•622905 


1 19 


-047478 


•676490 


20 


•036910 


•567146 


! 20 


•042055 


-628822 


20 


•047573 


•677358 


21 


•036993 


•568118 


1 21 


•042144 


•624789 


21 


•047668 


•678226 


22 


•037076 


•569089 


1 22 


•042233 


•625655 


22 


•047763 


•679093 


23 


•037159 


•570060 


23 


-042322 


•626570 


23 


•047859 


•679960 


24 


•037242 


•571030 


24 


•042412 


•627484 


, 24 


-047954 


•680827 


25 


•037325 


•571999 


25 


•042501 


•628398 


! 25 


•048050 


•681690 


28 


•037408 


•572967 


26 


•042590 


•629310 


! 26 


-048145 


•682553 


27 


•037492 


•573937 


27 


•042680 


•630222 


: 27 


-048241 


•683417 


28 


•037575 


•574899 


28 


•042770 


•631133 


1 28 


•048837 


•684280 


29 


•037658 


•575863 : 


29 


•042859 


•632043 


i 29 


•048433 


•685142 


30 


•037742 


•576827 


30 


•042949 


•632952 


i 30 


•048529 


•686002 


31 


•037826 


•577790 


31 


•043039 


•633861 


1 31 


-048625 


•686862 


32 


•037910 


•578752 


32 


-04S129 


•634768 


32 


-048722 


•687721 


33 


•037994 


•579713 


33 


-043219 


•635674 


33 


•048818 


•688581 


34 


•038078 


•580673 


34 


-0433C9 


•636580 


1 ^4 


•048915 


•689488 


35 


•038162 


•581631 


35 


•043400 


•637486 


i 35 


•C49011 


•090296 


36 


•038246 


•582589 


36 


•043490 


•638389 


1 36 


•C491C8 


•691153 


37 


•038331 


•583547 


37 


•043580 


•689292 


1 37 


•C492C5 


•692008 


38 


•038415 


•584503 


38 


-043671 


•640195 


i 38 


•049302 


•692863 


39 


•038500 


•585458 


39 


•043762 


•641096 


1 89 


•049899 


•693717 


40 


•038585 


•586412 


40 


•043S53 


•641997 


1 40 


•049296 


•694571 


41 


•038669 


•587365 


41 


•043943 


•642890 


41 


•049593 


•695424 


42 


•038754 


•588318 


42 


■044035 


•643796 


42 


•G49691 


•696276 


43 


•038839 


•589269 


43 


•044126 


•644694 


, 43 


•049788 


•097127 


44 


•038924 


•590219 


44 


•044217 


•645591 


44 


•049886 


•697978 


45 


•039009 


•591169 


45 


•044309 


•646488 


45 


•049988 


•698829 


46 


•039095 


•592117 


46 


-044400 


•647384 


46 


•050081 


•699677 


47 


•039181 


•593075 


47 


•044491 


•648272 


47 


•050179 


•700526 


48 


•039266 


•594012 


48 


•044583 


•649173 


48 


•C5C277 


•701878 


49 


•039351 


•594958 


49 


-044676 


•650076 


49 


•C5C376 


•702220 


50 


•039437 


•595902 


50 


-044767 


•650958 


50 


•040474 


•703066 


51 


•039523 


•596846 


51 


•044859 


•651850 


51 


•(;5r:572 


•708911 


52 


•039603 


•597789 


52 


•044951 


•652741 


52 


•050671 


•704757 


53 


•039695 


•59S731 


1 53 


•045043 


•653630 


53 


•050769 


•705600 


54 


•039781 


•599672 


i 54 


•045136 


•654520 


1 54 


•050868 


•706444 


55 


•039867 


•600612 


i 55 


•045228 


•655408 


55 


•050967 


•707287 


56 


•039953 


•601551 


1 56 


•045321 


•656296 


i 56 


•051006 


•708128 


57 


•040040 


•602489 


57 


-045413 


•657183 


57 


•051165 


•708969 


58 


•040126 


•603427 


i 58 


-045506 


•658069 


; 58 


•051264 


•709810 


59 


•043203 


-604263 


' 59 


-045599 


•658953 


i 59 


•051863 


•710649 


60 


•04C299 


-605299 


i 60 


•045692 


-659883 


: 60 


•051462 


•711489 



428 



Versed Sixes. 



18 DEGEEES. 1 


ID DEGEEES. 


20 DEGEEES. 




«... 1 


Niit No 


l...<a i;lun. ' 


Min. j 


Nat. No. 1 


Logarithm. 


i Min. , 


Nat. No. j 


Logariihm. 


1 





0-04S914 


8-689694 ! 


^' ! 


0-054481 


8-786248 


^ 1 


0-060308 


8^7S0371 




1 


'04dm : 


•690492 ! 


1 1 


•05-: '76 


•787003 


1 


•060407 


•7810S7 




2 


•049123 


•691283 ' 


2 : 


•054671 


-737753 


2 


•060507 


•7818:2 




3 


•049213 ■ 


•692084 


3 


•054766 


-733510 


3 


•060606 


•782517 




4 


•049304 


•692380 


4 1 


•054361 


*• -739263 


! 4 1 


•060706 


•783232 




5 


•049894 


•69C674 


5 ! 


•054956 


-740014 


1 ^ 


•060806 


•783945 




G 


•049434 


•694467 


6 1 


•055051 


•740766 


! 6 


•06C906 


•784659 




7 


•049575 


•695260 


^ ! 


•055146 


-741517 


7 


•061006 


•785371 




8 


•049 o85 


•696052 , 


8 1 


•vi55242 


•742266 


i 8 


•061106 


•786082 




9 


•049756 


•696343 ' 


9 


•055337 


•74:3015 


9 


•061206 


•736794 




10 


•049346 


•697033 ; 


10 


•055432 


•743764 


10 i 


•0613G6 


•787505 




11 


•049937 


•693423 


11 


•055527 


-744512 


11 ! 


•061407 


•788215 




12 


• -050028 ; 


•699213 


12 


•055623 


-745259 


12 i 


•061507 


•788924 




13 


•053119 1 


•700001 


13 


•055719 


•746005 


13 i 


•061608 


•789634 




It 


•050210 i 


•700788 


14 


•055S15 


•746752 


^ 14 ; 


•001708 


•790341 




15 


•050301 


•701576 


15 


•055911 


•747497 


15 


•0618C9 


•791049 




Ifi 


•050392 


•7G2361 


16 


•056007 


-748241 


i 16 


•061909 


•791756 




IT 


•050483 


•703147 


17 


•0561G3 


•743945 


17 


•062010 


-792463 




IS 


•050574 1 


•703931 


18 


•056199 


•749728 


18 


•062111 


•793169 




19 


•050666 : 


•704716 


19 


•056295 


•750472 


19 


•062212 


•793874 




20 


•050757 i 


•705498 


20 


•056S91 


•751213 


20 


•062313 


•794579 




21 


•050849 


•706232 ; 


21 


•056483 


•751955 


21 


-062414 


•795283 




1 22 


•050941 


•707063 


22 


•056531 


•752096 


22 


-062515 


•795937 




1 23 


•051032 


•707845 


23 


•050G81 


•753436 


23 


•062617 


•796693 




1 24 


•051124 ' 


•708624 


24 


•056777 


•754175 


i 24 


•062718 


•797393 




25 


•051216 ' 


•7C9404 


25 


•056874 


•754913 


i 25 


•062820 


•793C94 




26 


•051308 


•710183 ; 


26 


•056971 


•755652 


I 26 


•062921 


•793795 1 


27 


•051400 


•710961 


27 


•0570G8 


•756390 


• 27 


•063023 


•799496 1 


28 


•051492 


•711739 


23 


•057104 


•757126 


1 28 


•063124 


•800196 1 


29 


•051584 


•712516 ; 


29 


•057261 


-757862 


1 29 


•063226 


•800895 1 


30 


.•051676 


•713291 ! 


30 


•057358 


•758597 


SO 


•063828 


•801594 




31 


•051769 


•714067 . 


81 


•057456 


•759333 


! 31 


-063430 


•802293 




i 32 


•051861 


•714842 


32 


•057553 


•760067 


32 


•063532 


•802990 




1 33 


•051953 


•715615 


33 


•057650 


•760800 


33 


-003634 


•803683 




34 


•052046 


•716339 


34 


•057748 


-761534 


34 


•068736 


•804384 




35 


•052139 


•717162 


35 


•057345 


•762266 


85 


•063838 


•805080 




36 


•052232 


•717933 


36 


•057942 


•762992 


36 


•063949 


•805775 




i 37 


•052324 


•71 3704 


37 


•058040 


763728 


. 37 


•064043 


•806470 


3S 


•052417 


•719474 


35 


•058138 


•764459 


38 


•064145 


•807165 


, 39 


•052510 


•720244 


39 


•053236 


•765189 


i 39 


•064-248 


•S07853 


1 40 


•052603 


•721013 


40 


•058334 


-765918 


:' 40 


•064350 


•808551 


i 41 


•052697 


•7217S2 


41 


•058432 


•766647 


I 41 


•064453 


•8C9244 


i 42 


•052790 


•722549 


42 


•058530 


•767375 


; 42 


•064556 


•809937 


! 43 


•052S83 


•7-23316 


43 


•058623 


•768101 


1 43 


•064659 


•810628 




44 


•052976 


•724082 


44 


•058726 


•76S829 


; 44 


•064762 


-811319 




45 


•053070 


•724848 


45 


•058324 


•769555 


i 45 


•064865 


•8120(0 




; 46 


•053163 


•7-25613 


46 


•058922 


•770279 


46 


•064968 


•312G93 




47 


•053257 


•726377 


47 


•059ik21 


-771005 


47 


•005071 


•8133S7 


' 


48 


•053351 


•7-27140 


48 


•059119 


77172D 


:• 48 


•065174 


•814076 




49 


•053444 


•727903 


49 


• •059218 


772453 


' 49 


•065278 


.•814705 


50 


•053538 


•7-23666 


50 


•059316 


773175 


\: 50 


•065381 


•S15452 


51 


•053632 


•729427 


51 


•059415 


! -773897 


i 51 


-065485 


. •SIGISO 




52 


•053726 


•730183 


52 


•059514 


•774619 


52 


•065558 


. •816325 




53 


•C53820 


•73G947 


53 


•059613 


•775340 


53 


•065692 


. -817511 




54 


•053915 


- •731706 


54 


•059712 


•776060 


i 54 


•065796 


. •81819G 




55 


•054909 


•732465 


55 


•059311 


776780 


W 55 


•065899 


•818831 


56 


•054103 


! -733224 


56 


•059910 


777500 


ii 56 


•066003 


. -819565 




57 


•054193 


1 •733931 


57 


•060009 


i •778218 


57 


; -OOGirj 


•823240 




58 


•054292 


• ^734737 


53 


•060109 


1 •778936 


11 58 


i •066-211 


•82C931 




59 


•054387 


1 •73r)493 


59 


•060208 


; -779653 


' 59 


•066315 


•821614 


j 


60 

[. . . 


•054481 


1 •736243 


60 


-060308 


•783371 


' 60 


•066420 


•822296 


i 









Ex 


FERXAL ? 


EC ANTS. 






429 


18 DEGREES. 


19 DEGREES. 


20 DEGREES. 


Mill. 1 N:i(. No. 


Lo^ari.hn,. | 


mu. 


Nat No. 


Lr.iraiidim 1 


Min. 


"n.:n:" 


Logarithm 





051462 


8-7114S9 1 





0^057621 


8-700573 





0^064178 


8^8073S5 


1 


051562 


•712327 


1 


•057727 


•761376 


1 


•064-29;) 


•808147 


2 


C51661 


•713164 


2 


'057833 


•762174 


2 


•0644(J3 


•8089C8 


8 


051761 


•714001 


8 


•057939 


•762071 


3 


•064-511 


•8G9669 


4 


051861 


•714S3S 


4 


•C58045 


•763767 


4 


•0646:, 9 


•810430 


5 


051960 


•715673 1 


5 


•G5S152 


•764562 


5 


•064743 


•811190 


6 


052060 


•7165C8 1 


6 


•058258 


•765358 


6 


•064856 


•811950 


7 


052161 


•717342 


7 


•058365 


•766152 


7 


•064 969 


•812708 


8 


052261 


•718175 1 


8 


•058472 


•766945 


8 


•C6.;i83 


•813465 


9 • 


052361 


•719 J( 8 1 


9 


•05S579 


•767738 


9 


•C6;197 


•S14224 


10 


052461 


•719839 I 


10 


•058686 


•768531 


10 


•065310 


•814981 


11 • 


052562 


•72.671 i 


11 


•058793 


•769323 


11 


•065424 


•815737 


12 ' 


052663 


•7215G2 1 


12 


•058900 


'770114 


12 


'065538 


•816493 


18 


052763 


-722332 i 


13 


•059307 


•77^904 


13 


•065652 


•817249 


14 ' 


052864 


•723160 1 


14 


•059115 


•771695 


14 


•065766 


•818003 


15 


052965 


•72399) ! 


15 


•059222 


•772484 


15 


•0658S1 


•818758 


16 ' 


053066 


•724S17 


16 


•059330 


•773272 


16 


•065995 


•S19511 


17 • 


053167 


•725644 


17 


•C59438 


•774060 


^17 


•066110 


•82C265 


18 


053268 


•726470 


18 


•059545 


•774848 


18 


•066224 


•821018 


19 


053370 


•727297 i 


19 


•059654 


•775636 


19 


•0663.'.-9 


•821769 


20 ' 


G53471 


•7L8I22 ^ 


20 


•C597G2 


•776421 


2>) 


•066454 


•822521 


21 


053573 


•72S947 i 


21 


•059870 


•777207 


21 


•066569 


•823272 


22 


053675 


•729770 : 


22 


•059978 


•777993 


22 


•066684 


•824023 


23 


053777 


•730594 \ 


23 


•060087 


•778777 


23 


•0668:)() 


•824773 


24 


053879 


•731415 1 


24 


•060195 


•779561 


24 


•066915 


'825523 


25 


053981 


•732237 • 


25 


•060304 


•7SC343 


25 


•067030 


•S26271 


2Q 


054083 


•733C58 i 


26 


•060412 


•781127 


26 


•067146 


•827G1D 


27 


054185 


•733878 , 


27 


•060521 


•7819:9 


27 


•067262 


•827767 


28 


054287 


•734698 ! 


28 


•060630 


•7S2C90 


28 


•067377 


•828514 


29 


054390 


•735517 i 


29 


•060740 


•7S3471 


29 


•067493 


•829260 


80 


054492 


•736335 


80 


•060849 


•784252 


80 


•067609 


•8300C7 


31 


054595 


•737153 ' 


81 


•06(958 


•785031 


81 


•067726 


•830753 


32 


054G98 


•737970 i 


32 


•061068 


♦7S5810 


82 


•067842 


•831497 


38 


054801 


•738785 i 


38 


•061177 


•786588 


33 


•067958 


•832242 


34 


0549D4 


•734602 ' 


84 


•061287 


•7S7367 


84 


•068075 


•8329S3 


85 


055007 


•740417 


35 


•061397 


•788144 


85 


•068191 


•833729 


36 


055110 


•741231 


36 


•061506 


•7SS915 


86 


•0683C8 


•834472 


87 


055213 


•742044 


37 


•0G1G16 


•7S9G96 


87 


•068425 


•835214 


88 


055317 


•742S57 


38 


•061726 


•79.0472 


88 


•068542 


•835957 


89 


055420 


•743670 


89 


•061837 


•791247 


89 


•068669 


•836697 


40 


C55524 


•744482 


40 


•0G1947 


•792021 


40 


•068775 


•837430 


41 


055628 


•745293 


41 


•062058 


•792795 


41 


•06SS93 


•838178 


42 


055732 


•746103 


42 


•062168 


•793568 


42 


•0(:9.ii 


•838919 


43 


055836 


•74G912 


48 


•062379 


•794340 


43 


•069129 


•839658 


44 


055940 


•747721 


44 


•062390 


•795113 


44 


•069247 


•840397 


45 


056044 


•748530 


45 


•062501 


•7958S4 


45 


•069364 


-.841 135 


46 


056148 


•749338 


46 


•062612 


•796654 


46 


•069482 


•841871 


47 


•056253 


•750145 


47 


•062723 


•797424 


47 


•069600 


•8-i2608 


48 


•056357 


•75C951 


4.8 


•062834 


•798197 


48 


•069718 


•84:^345 


49 


•056462 


•751757 


49 


•062945 


•798964 


49 


•069S36 


•844082 


50 


056567 


•752563 


50 


•063057 


•799731 


50 


•069955 


•844817 


51 


•056072 


•753367 


51 


•063168 


•800499 


51 


•070073 


•845553 


52 


•056777 


•754171 


52 


•063280 


•801267 


52 


•070192 


•846287 


; 53 


•056882 


•75-1978 


53 


•063392 


•802033 


53 


•070311 


•847021 


54 


•056987 


•755776 


54 


•063504 


•8^2799 


54 


•070480 


•847754 


, 55 


•057092 


•75G578 


55 


•0G3616 


•8;3565 


55 


•070549 


•848487 


56 


•057198 


•757380 


56 


•0G3728 


•804331 


56 


•070668 


•849-220 


57 


•057304 


•758181 


57 


•(i63S40 


•8 .15094 


57 


•070787 


•849952 


58 


•0574C9 


•758980 


58 


•06.'.953 


•805858 


58 


•07C9;)6 


•850682 


59 


•057515 


•750770 


59 


•064065 


•806621 


59 


•071(;25 


•851414 


60 


•057621 


•760578 


60 


•0(m78 


•807385 


60 


•071145 


•852144 



430 



Verskd Sines. 



21 DEGPwEES. 1 


22 DEGEEES. 


23 DEGEEES. 


Mil:. ; 


Na'. No. 


Ln.^ii.hm. 1 


Mi... ! 


Ni^t. No 


[...uarilhir. i Min. j 


Nat. No. '■ 


I,..;:aritlm. 





0-066420 


S-S22296 i 


i 


0-072816 ' 


8-862227 


C-079498 


8 900340 


1 


•066524 ' 


•82-2977 


1 


•07-2925 


•862877 ; 1 


•079609 


•90f9C2 


2 


•066628 


•823658 


2 1 


•073034 


•863526 : 2 


•079723 ; 


•901582 


3 


•066733 


•824388 


3 1 


•073143 


•864175 1 


3 


•079837 i 


•9G2201 


4 


•066837 ; 


•825lil8 


4 i 


•078253 


-864823 1 


4 


•079951 ' 


•902821 


5 


•066942 1 


•825697 


5 i 


•073862 


•865471 i 


5 


-080064 


•9;;3480 


C 


•067047 


•826376 i 


6 1 


•073471 


•866118 i 6 


•C80178 


•904057 


T 


•007151 


•S27C54 


7 i 


•073581 


•866765 i| T 


'08(3293 


•904675 


8 


•007256 


•827731 1 


8 ! 


•073690 


-867411 1 8 


♦C8C407 


.905293 





•067361 


•3284(9 1 


9 i 


•078800 


•868056 3 9 


•C80521 


•905910 


10 


•C67466 


•829085 : 


10 i 


•073910 


•868701 ii 10 


•C80686 


•906527 


11 


•067571 


•829760 : 


11 ; 


•074020 


•869346 ' 11 


•0SGT50 


•907143 


13 


-067676 


•S3i^436 ; 


12 i 


•074130 


•869991 - 12 


•080865 


•907758 


13 


•G67781 


•831110 ■ 


13 : 


•074239 


•8706:34 i 13 


•08 979 


•908374 


14 


•067837 


•831785 1 


14 • 


•074349 


•871277 1 14 


•081' 94 


•9:8988 


35 


•067992 


.832459 . 


15 : 


•074460 


-871920 15 


•081209 


•9-9603 


16 


•063097 


•833131 


16 I 


•074570 


•872562 ! 16 


•081824 


•91C216 


11 


•068203 


•833804 


17 ! 


•074680 


•878203 IT 


•G81489 


•^10830 


18 


•0683;:9 


•8:34476 


18 i 


•074790 


•873845 18 


T81554 


.911443 


19 


•068415 


•835148 


19 


•074931 


•874486 , 19 


•081669 


•912056 , 


2;) 


•068520 


•835819 


20 


•075011 


•875126 ;| 20 


•081784 


-912668 \ 


21 


•068626 


-836489 


21 


•075122 


'875766 ,. 21 


•C81899 


•913279 


22 


•068732 


•837159 


22 


'075282 


•876405 ii 22 


•C82014 


■913890 


23 


•068838 


•837829 


23 


•075343 


•877044 i 28 


•082130 


•914501 


24 


•0C8944 


•838497 


24 


•075454 


•877682 ■ 24 


•082245 


•915111 


25 


•0C9 50 


•839165 


25 


•C75565 


•878320 :: 25 


•082361 


•915721 


26 


•:C9157 


•839833 


26 


•075076 


'876957 ;i 26 


•08-2476 


•916:381 


27 


•C692G3 


•840501 


27 


•075787 


•879594 ;| 27 


•0S2592 


-916989 


2S 


•C69:3G9 


•841167 


23 


•075S98 


'880280 :! 28 


■082708 


-917548 


20 


•CG9476 


•S41834 


29 


•0760r9 


•88C866 ;1 29 


•082824 


•918156 


. 80 


•0C9582 


•842199 


30 


•C76121 


'8S15G2 I 


30 


•082940 


-^18764 


SI 


•369359 


•84;3165 


31 


•076232 


•882137 1 


31 


•083056 


•919371 


S2 


•0G979G 


•848829 


82 


•076343 


•882770 ; 


82 


•083172 


-919971 


i S3 


•0C99.3 


•844493 


33 


•076455 


'883405 i 83 


•088288 


•920583 


i 34 


•070010 


•845157 


34 


•076566 


-884':^88 1 34 


•C83404 


•9-21189 


i 35 


•370117 


•S4:G30 


35 


•v76678 


•834670 J 35 


•08:3521 


•^21795 


1 36 


•:7^:^224 


•516483 


36 


•076790 


•885303 36 


•083637 


•922400 


i 87 


•'T.GSl 


•847145 


3T 


•070902 


•8S5935 37 


•083754 


•923004 


• ss 


• -■ I Jioi 


•8478^5 


38 


•077014 


•886567 38 


•083871 


•9236G9 


; S9 


•:70545 


•848467 


39 


•077125 


•837197 i 39 


•083987 


•924212 


; 40 


•G7C653 


•S49127 


40 


•077237 


•837828 1 40 


•084104 


•924815 


■ 41 


•07076a 


•549787 


41 


•077350 


'88S458 ;! 41 


•C84221 


•925418 


• 42 


•07C837 


•850446 


42 


•077562 


•889088 


42 


•084337 


•926020 


43 


•:.7G075 


: •851106 


43 


•077574 


•889717 


43 


•084454 


•926623 


AJ- 


•371083 


! ^851764 


44 


•077687 


•890346 


44 


•084572 


-927224 


45 


•07119) 


; •852122 


45 


•077799 


-890974 


! 45 


•084689 


•927824 


46 


•G71293 


•853079 


46 


•€77912 


•8916<32 


46 


•084806 


•928^25 


: 47 


■ •371406 


•85:3735 


47 


•078024 


•892229 


47 


•084923 


-929025 


48 


•371514 


•854301 


48 


•078187 


■892S53 


48 


•085040 


•929625 


49 


•C71622 


•855048 


49 


•078250 


; -893482 


49 


•085158 


•9302-24 


50 


•371730 


•855703 


' 50 


•078363 


: -894108 


50 


•085275 


•930823 


51 


; -371339 


•856358 


; 51 


•078476 


: •894734 


51 


•085893 


•931422 


. 52 


•071047 


1 -857012 


52 


•0785S9 


•895858 


52 


•085510 


•982019 


53 


•072C55 


1 -857665 


1 53 


•C7S7C2 


•895983 i 53 


•085628 


-982617 


' 54 


•0721G4 


! •85.8319 


i 54 


•078815 


; -896607 54 


i -085746 


•988214 


55 


•072272 


1 •35S972 


!i 55 


•07S928 


-897230 1 55 


•085864 


•933811 


56 


•072381 


i •859G24 


: 56 


•079341 


: ^897853 ! 56 


t -085982 


i -934407 


■ 57 


•07249.) 


i •860275 


, 57 


•070154 


-S9S475 ! 57 


i -086100 


i •935003 


58 


■•072593 


■ •8a'93T 


1 58 


•079268 


-899097 


58 


, •086218 


: -935598 


59 


•0727.37 


: •831578 


1 59 


•079382 


•899719 


59 


•086336 


1 -936193 


63 


•072816 


1 •S62-22'r 


1 00 


•079495 


1 -900340 


! 60 

i 


•0864M 


•93678? 



External Si:cants. 



431 



1 


21 DEGREES. 


1 


22 DEGREES. 


2 


3 DEGREES. 


Mill 


Nat. No. 


L.,-...itl.;n. 


j Min. 


Nut. No. 


I-osari.l.m. 


Min. 


Nat. No. 


Logaiiihm. 





0-071145 


8-852144 





0^078535 


8-895061 





G-.8G360 


8-93G314 


1 


•071265 


•852374 


i 1 


•0783G2 


•895762 


1 


•086495 


•936993 


2 


•0T1384 


•853603 


i 2 


•078789 


•896462 


2 


•08G629 


•937663 


3 


•071504 


•854362 


i 3 


•078916 


•8971G2 


3 


•086763 


•933336 


4 


•071624 


•855361 


1 4 


'079343 


'897S62 


4 


•086898 


•939310 


5 


•071744 


•855788 


5 


•079170 


•898561 


5 


•087033 


•939682 


6 


•071835 


•856516 


6 


•079297 


•899259 


6 


•C87167 


•940353 


T 


'071935 


•857243 


7 


•079425 


•899957 


7 


•087302 


•941025 


8 


•072105 


•857969 


8 


•079553 


•930655 


8 


•087437 


•941697 


9 


•072226 


•858695 


9 


•079683 


•931351 


9 


•087573 


•942368 


10 


•072347 


•859423 


10 


•0798)3 


•9.2143 


10 


•087708 


•943040 


11 


•0724G8 


•860144 


11 


•079933 


•932744 


11 


•087843 


•943709 


12 


•072539 


•860869 


12 


•080065 


•903-441 


12 


•087979 


•944:379 


13 


•072710 


•861592 


13 


•080193 


•904135 


13 


•C88115 


•945049 


14 


•072S31 


•862316 


14 


•083321 


•904833 


14 


•083251 


•945717 


15 


•072952 


•863039 


15 


•083453 


•9 5525 


15 


•C833S7 


•946386 • 


IG 


•073374 


•833761 


16 


•08j5T8 


•9G6218 


16 


•088522 


•947C53 '• 


IT 


•073195 


•884433 


17 


•O807O7 


•936911 


^17 


•083659 


•947722 , 


18 


•073317 


•885204 


18 


•C8,:8'J6 


•937605 


18 


•083795 


•948389 ' 


19 


•073439 


•865925 


19 


•080965 


•908298 


19 


•08S932 


•949357 ' 


2J 


•073531 


•866646 


23 


•0S1C94 


•938990 


20 


•089368 


•949723 


21 


•073683 


•867365 


21 


•081223 


•9u9G81 


21 


•C89205 


•9533S9 


22 


•0738 J5 


•868088 


22 


•081353 


•910372 


22 


•089312 


•951054 


23 


•073927 


•838334 


23 


•081482 


•911063 


23 


•0S9479 


•951720 


24 


•074349 


•869521 


24 


•G81612 


•911754 


24 


•G89316 


•952384 


25 


•074172 


•870239 


25 


•081742 


•912444 


25 


•089753 


•953049 


23 


•074294 


•87C956 


26 


•081872 


•913133 


26 


•C89S91 


•953714 


27 


•074417 


•871674 


27 


•082302 


•918822 


27 


•093023 


•954377 1 


23 


•074543 


•872390 


28 


•082132 


•914510 


28 


•090165 


•955040 


29 


•074663 


•873106 


29 


•082262 


•915193 


29 


'093303 


•955703 


30 


•074787 


■873829 


30 


•082392 


•915SS7 


30 


•093441 


•956366 


31 


•074913 


•874537 


31 


•082523 


•916574 


31 


•090579 


•957G28 


32 


•075333 


•875251 


32 


•082653 


-917260 


82 


•C90717 


•957689 


33 


•075158 


•875965 


33 


•082784 


•917947 


83 


•C93855 


•953350 


34 


•07528) 


•876678 


34 


•082915 


•918632 


34 


•093994 


•959311 


35 


•075404 


•877391 


35 


•083U46 


•919317 


35 


•091132 


•959372 


36 


•075527 


•878103 


36 


•083177 


•920002 


86 


•C91271 


•96C333 


37 


•075651 


•878316 


37 


•083308 


•920687 


37 


•C91410 


•96; 992 


38 


•075775 


•879527 


38 


•083443 


•921372 


38 


•091553 


•961652 


39 


•075930 


•8S3239 


39 


•083570 


•922054 


39 


•G91GS3 


•9G2310 


40 


•076324 


•8S;949 


40 


•083702 


•922738 


40 


•C91827 


•962969 


41 


•076143 


•8B1659 


41 


•083834 


•923421 


41 


•C919G6 


•963627 


42 


•076273 


•83-2368 


42 


•083966 


•9-24104 


42 


•C92105 


•964285 


43 


•076393 


•883379 


43 


•084093 


•924786 


43 


•092245 


-964942 


44 


•076522 


•883787 


44 


•084230 


•925467 


44 


•C92385 


•965600 


45 


•076647 


•884-495 1 


45 


•084362 


•92G148 


45 


•C925-24 


•966255 


46 


•076772 


•885203 1 


46 


•084495 


•926829 


46 


•G92664 


-9G69I2 


47 


•076S97 


•8S5939 


47 


•084627 


•927510 


47 


•C92834 


•967567 


48 


•077022 


•8S6616 


48 


•084760 


•928190 


48 


•C92944 


•96S223 


49 


•077148 


•837323 


49 


•084893 


•928869 


49 


•093085 


•96S87S 


50 


•077273 


•888329 


50 


•085U25 


•929548 


50 


•193225 


•969532 


51 


•077399 


•883734 


51 


•085158 


•930227 


51 


•C933G6 


•970187 


52 


•077525 


•8894:39 


52 


•085291 


•93':904 ' 


52 


•093506 


•970840 


53 


•07765') 


•893143 


53 


•085425 


•931583 i 


53 


•C93G47 


•971494 


54 


•077776 


•890848 


54 


•085558 


-932260 ' 


54 


•093783 


•972147 


55 


•077933 


•891551 


55 


•085691 


■932636 


55 


•093029 


•972300 


56 


•078329 


•892254 


56 


•085825 


•933613 


56 


•094;)70 


•973452 


57 


•078155 


•S9295(i 


57 


•085958 


•934288 


57 


•(94212 


•974104 


5S 


•078282 


•893659 


53 


•086392 


•934964 


58 


•C94353 


-974755 


59 


•078108 


•891361 


59 


•086225 


•935631 


59 


•C94495 


•975407 


6:) 


•078535 


•895061 


63 


•086360 


•936814 


60 


•G94636 


•976057 



432 








Versed 


Sines. 










24 DEGEEES. | 


25 DEGEEES. 


26 DEGEEES. 


Min. 


1 Nat. No. 


; Losari:l,,u. 1 


Min 


Nat. Xc. 


! LoiraritLm. 


j Min. 

1 


j Nat. No. 

0-101206 


Logarithm. 





! C-0S(>4,54 


8-936787 j 





0^C93092 


; 8-9717C3 


i 9-005206 


1 


! -086578 


•937882 i 


1 


; •093815 


; -972273 


; 1 


•101333 


j -005758 


2 


: -086695 


■ -937975 


2 


. -093939 


i •972843 


^i 2 


•101461 


•006800 


S 


1 -os^sio 


•9CGoC9 


3 


•094061 


1 ^978411 


i 3 


•101588 


•006843 


4 


"! -086929 


: :939162 


4 


; •094184 


j -978979 


■' 4 


' ^101716 


•007892 


5 


! -087047 


•939754 


5 


: -094308 


j -974547 


i 5 


1 -101845 


! •007938 


6 


1 -087166 


•94^;346 


6 


i -094481 


i •975115 


6 


i •101978 


-00848p 
-00932T 


7 


' -087286 


•94:938 


7 


i -004555 


, -975683 


; 7 


: -102100 


8 


i -087404 


-941529 


8 


: -094678 


•976250 


: 8 


•1 2229 


•009572 


9 


■ -087523 


•912123 


9 


•094S'2 


i -976816 


9 


•1 2357 


•010116 


10 


•0S7642 


•942711 


13 


: -094925 


•977S83 


10 


, •1.24S5 


•010660 


11 


•087761 


-913800 


11 


' •C95J49 


■ •977948 


: 11 


•1.2318 


•011204 


12 


•0S78S0 


•943SS9 


12 


•095173 


: ^978514 


i 12 


•102 743 


•011746 


18 


•0S7999 


•944478 


18 


; ^095297 


: •979378 


i 18 


-102870 


•012289 


14 


•08S119 


•945067 


14 


i -095421 


i -979643 


14 


-102999 


•012832 


]5 


•088288 


•945656 


15 


' ^095545 


! •98 207 


15 


-108128 


-018373 


1() 


•OSS^jS 


•9462^8 


16 


•095669 


•983771 


16 


•103256 


•018915 


17 


•088477 


•940833 


17 


•395793 


•931334 


i 17 


-108835 


-014456 


IS 


•088597 


•947418 


IS 


•C93913 


-931S9S 


j IS 


•103514 


•014998 


19 


•G8S716 


•943004 


19 


•C96342 


•932460 


! 19 


•103648 


•015588 , 


2'J 


: -OBSS^e 


•918593 


2) 


•095166 


•938;:23 


1 20 


-103772 


•016078 


21 


-088956 


•949175 


21 


•096291 


•938585 


! 21 


-108931 


■016618 


22 


•0S9.J76 


•949761 


22 


-096415 


•934146 


22 


-104030 


•017157 


23 


•C89196 


•953346 


28 


•096540 


•934707 


23 


•104159 


•017696 


24 


•089816 . 


•950981 


24 


-096665 


-985208 


24 


-104238 


•018285 


25 


•089437 


•951515 


25 


•096793 


•935329 


25 


-104417 


•018778 


23 


•089557 


•952^98 


26 


•096914 


•9838S8 


28 


•104547 


•019811 


27 


-C89677 : 


•9526S1 


27 


•097O40 


-936948 


27 


•104676 


-019847 


2S 


-089798 .: 


•953265 


2S 


•0971 04 


•937507 


28 


•104806 


-020887 


29 


•069918 ^ 


•95884S 


29 


•097293 


•933306 


29 


-104936 


•020924 


30 


•C9J039 


•954429 


SO 


•097415 


•988624 


30 


-105066 


•C21460 


31 


-09:;159 


•955011 


31 


•09754) 


•939182 


31 


•105196 


•021997 


32 


-09328;) . 


•955592 


32 


•097666 


■939741 


32 


•105326 


•022583 


33 


-090431 ' 


•956173 


33 


-097791 


•993293 


33 


•105456 


•023069 . 


M 


-090522 1 


•956753 


84 


-097916 


•990854 


34 


•105586 


•023603 V 


35 


•090648 1 


•957833 


35 


-098342 


•991411 


35 


-105716 


•024189 ,> 


36 


'090764 i 


•957918 


86 


•09816S 


•991968 : 


36 


•105S46 


•024673 , 


3T 


'09 3885 ; 


•95S492 : 


87 


•C9S298 


•992524 


37 


-105977 


•025210 v 


38 


-C91006 : 


•959071 ' 


38 


-C9S419 


•998079 i 


38 


•106107 


-025742 


39 


•091127 1 


•959649 : 


39 


•098545 


•998634 ' 


39 


•106287 


•026275 


40 


•091249 


^6;228 


40 


•098671 


•994189 


40 


•106367 


•026808 


41 


•091370 


'960805 ; 


41 


. -093797 


•994743 


41 


•106493 


-027842 N 


42 


•G91492 


•931382 


42 


-098928 : 


•995297 


42 


•106629 


•037874 


43 


•091613 


•961959 


43 


•099 )49 ! 


•995851 


43 


-106760 


•C28406 


44 


•C91785 ' 


•962585 i 


44 


•099175 i 


•996404 


44 


-106890 


•028988 . 


45 


•C91357 ■ 


•968111 


45 


•099302 


•993957 


45 


•107021 


•029470 ■. 1 


46 


•091979 1 


•9686ST 


■m 


-099428 


•9975C9 


46 


•107152 


•030000 . f 


47 


•092101 i 


•964262 


47 ■. 


•099555 


•993061 , 


47 


•107288 


-030531 ! 


48 


•G92223 ! 


^964836 


48 1 


'099681 i 


•993613 


48 


•107414 


•031061 


49 


'0^2345 • 


•965412 


^ : 


•099308 i 


•999164 


49 


•107.545 


•031592 


50 


•092467 


•$65985 


50 


•0999:34 i 


•999715 


50 


-107677 \ 


-032121 


51 


-092589 


•966559 


51 1 


-100061 , 


9 000-266 


51 i 


•1078G8 1 


•032651 i 


52 


•092711 ' 


•967181 


52 j 


•10018S 1 


•00C817 ' 


52 


-107940 i 


•03318f) 1 


53 


•092S34 ■ 


•967705 


53 : 


-100315 i 


•001366 ■ 


53 ; 


•108371 j 


•033709 


54 


•092356 


•968277 


54 i 


•100442 1 


•001916 


54 = 


-10S232 i 


•034237 


55 


-098378 ; 


•968849 


55 ; 


•100570 i 


'002465 , 


55 1 


•108884 ! 


•034765 


56 


•0982n ' 


•9-39 121 


56 


•100697 i 


^003014 i 


56 1 


•108466 


•035293 


57 


•093324 ' 


•9C9991 


57 , 


•100S24 i 


'0G3563 


57 1 


•108593 


•035820 


58 


•0934-17 i 


:970533 ii 


58 i 


•100951 i 


•004111 


58 i 


•108730 


•036348 


59 


-093560 ; 


•971133 1 


59 


•101079 


•004660 


59 i 


•108862 


•036874 


G'} 


•093693 ; 


'^71703 1 


.. 1 


•1^206 


•005206 


60 


-108994 


•037401 









External 


Secants. 






433 


I 24 DEGREES. 


j 25 DEGREES. 


! 

1 

; Mm. 




26 DEGR] 

Nat No. 


EES. 


Min. 


Nat. No. 


1 Lo^arilh-.n. 


j! Mill. 


' Nal. No. 


1 L..ga,ithtn. 


Logarilliiii. 





0-094G3G 


8-976057 


,' ^ 


, 0-103378 


• 9-014427 


0-1126C2 


9 051546 


1 


•C94778 


•976708 


r 1 


•103523 


! •015056 


i 1 


•112760 


•C52154 


2 


•C94920 


•977357 


1: 2 


•103678 


1 •015685 


! 2 


•112918 


-052763 


3 


•C950G2 


•9780G8 


ll 3 


•103828 


j -016312 


1 3 


•113076 


•053370 


4 


•G95204 


•978657 


i 4 


•103977 


•016939 


4 


•113235 


-(58979 


5 


•C95347 


•979306 


i 5 


•104123 


•017566 


5 


•113393 


•C54586 


6 


•C95489 


•979954 


6 


•104279 


•018194 


6 


•113552 


-C55195 


7 


-095632 


•980603 


'^ 


•104429 


•018821 


7 


•113710 


-C55799 


8 


•C95775 


•981250 


1^ 8 


•104530 


•019-447 


8 


•113869 


•056406 


9 


•095918 


, -981898 


ii 9 


•104730 


•020072 


9 


•114028 


•057012 


10 


•C96061 


•932546 


1^ 10 


-104831 


•020696 


10 


i -114187 


•C57618 


11 


•096204 


•933191 


!' 11 


•105032 


•021323 


11 


1 -114347 


-058224 


12 


•C9G847 


•933837 


' 12 


•105184 


•021948 


12 


•1145G6 


•C58S28 


13 


•096490 


•984483 


; 13 


•105335 


•022572 


13 


•114666 


•059434 


14 


•C96G34 


•985129 


1 14 


•105486 


•G23196 


14 


•114826 


•060(89 


15 


•C9G777 


•935774 


15 


•105638 


•023820 


\ IS 


•114985 


•060642 


10 


•096921 


•936417 


16 


•105790 


•024444 


Ul6 


•115145 


•C61246 


17 


•C97065 


•937062 


17 


•105942 


•025066 


• 17 


•115306 


•061850 


18 


•0972Q9 


•937707 


i 18 


•106C94 


•025690 


: 18 


1 ^115460 


•062454 


19 


•G97353 


•9S8350 


19 


-100246 


•026312 


19 


•115626 


•C63C57 


20 


•C97493 


•988994 


20 


-106898 


•026934 


' 20 


•115787 


•063660 


21 


'C97642 


■989636 


21 


-106551 


•027556 


' 21 


■115948 


•064262 


22 


•C97737 


•990279 


: 22 


-106703 


•028177 


22 


•1161(8 


•064863 


23 


•C9r931 


•99;;92l 


, 23 


•106856 


•028798 


23 


•1162C9 


•C 65465 


24 


•C9S076 


•991563 


i 24 


•107009 


•029419 


24 


-116431 


•066067 


25 


•C98221 


•992205 


25 


•107162 


•030040 


25 


-116592 


•066667 


26 


•C98366 


•992345 


26 


•107315 


•030659 


26 


•116753 


•C67268 


27 


•C98511 


•993486 


: 27 


•107468 


•031279 


27 


-11G915 


•067869 


28 


•r986c7 


•994127 


28 


•107621 


•031698 


28 


•117C77 


•C 68470 


29 


•(98802 


•994766 


' 29 


•107775 


•032518 


29 


•1172S9 


•C69070 


30 1 


•r98948 


•995406 


^ 30 


•107929 


•033136 


30 


•117400 


-C696C9 


81 


•C 99094 


•996046 


31 


•108C82 


-033754 


31 


•117562 


-070269 


32 


•C 99240 


•993634 


, 32 


•108236 


•034373 


82 


•117725 


-070868 


33 


•C99386 


•997323 


1 33 


•10S390 


•034991 


33 


•117888 


•071467 


34 


•C99532 


•997961 


: 34 


•10S544 


-035607 


34 


•118049 


•072064 


35 


•C99G78 


•998599 


: 35 


•108699 


•036225 


35 


•118212 


•072663 


36 


•C99824 


•999236 


i 36 


•108854 


•036842 


36 


•118875 


•C73261 


37 


•099971 


•999873 


! 87 


•1C90C8 


•037459 


: 37 


•118539 


•078861 


38 


•100118 


9-000510 


38 


•109163 


•038074 


i 88 


•118702 


•074456 


39 


•100264 


•001146 


39 


•109318 


•038690 


89 


•118865 


•075053 


40 


•100411 


•001733 


. 40 


•1G9473 


•08930G 


40 


•119028 


•075640 


41 


•100558 


•002418 


! 41 


•109628 


•039920 


: 41 


•119192 


•076246 


42 


•100706 


•003053 


: 42 


•109783 


•040535 


42 


•119355 


•076842 


43 


•100853 


•003688 


43 


•109989 


•041150 


43 


•119519 


•077438 


44 


•101000 


-004322 


44 


•110094 


•041764 


44 


•119C83 


•078033' 


45 


•101148 


•004957 


45 


•110250 


•042378 


45 


•119848 


•078629 


46 


•101296 


-005591 


: 46 


•110406 


•042991 


46 


•120012 


•079222 


47 


•101444 


•006224 


, 47 


•110561 


•043604 


' 47 


•120176 


•079817 


48 


•101591 


•006857 


48 


•110717 


•044217 


48 


•120340 


•080412 


49 


•101740 


•007491 


49 


•110874 


•044829 


49 


•120505 


•081006 


50 


•101835 


•008122 


50 


•111030 


•045441 


50 


•120670 


•081599 


51 


•102.)37 


•008755 


51 


•111187 


•046053 


51 


•120835 


•082198 


52 


•102185 


•009385 


52 


•111344 


•046665 


52 


•121000 


•082786 


53 


•102334 


-010018 


; 53 


•111500 


•047276 


53 


•121166 


•C88879 


54 


•102482 


•010649 


, 54 


-111G57 


•047887 


54 


-121331 


•088971 


55 


•102631 


•011279 


1 55 


•111814 


•04849T 


55 


-121498 i 


•08456S 


56 


•102780 


•011910 


56 


•111972 


•C49108 


50 


-121662 


•C85155 


57 


•102930 


•012539 


57 


-112129 


•019718 


57 


•121828 


'085740 


58 


•103079 


•013170 


58 


-112287 


•05032S 


58 


-121994 


•086338 


69 


•103223 


•U1379S 


59 


•112445 


•050938 


59 


-122160 


•C8C929 


60 


•103878 


•014427 


00 


•112602 


-051546 


60 


122327 


•087520 



4 '^4 


57 DEGE] 




Versed Sines. 










EES j 


28 DEGREES. 


29 DEGEEES. 


«•:- 


N-t N'; 


T.oeanlhm 


Min. 


Nat No ! 


I.ogaritl m. 


Min. 


Nat No 


Logaiilhm. 


i) 


-108994 


9-037401 1 





0117052 


9-068380 





0-125380 


9-(9S229 


1 


109126 


037927 : 


1 


•117189 i 


•068887 


1 


•125522 


•(98717 


2 


•109258 


•038452 i 


2 ' 


•117326 : 


•069393 


i 2 


•125663 


•099206 


3 


•109390 


•038978 


3 


•117462 ! 


•C69899 


3 , 


•125S04 


•099693 


4 


•109522 


•039502 


4 


•117599 ! 


•070404 


4 


•125945 


•10019>. 


5 


•1G9655 


•040027 


5 ' 


•117736 1 


•07C910 


5 


•126(86 


•100663 


6 


•109787 


•C40551 


6 


•117873 


•071415 


6 


•126228 


•101155 


T 


•109920 


•041076 


7 


•118010 


•071919 


7 


•126370 


•101642 


8 


•110053 


•041600 


8 


•118147 


•07-2424 


8 


•126512 


102129 


9 


. -110185 


•042123 


9 : 


•1132S5 


•072928 


! 9 


•126653 


•102614 


10 


•110318 


•C42645 


10 ■ 


•118422 


•073432 


1 10 


•126794 


•103100 


11 


•11C451 


•G43168 


n ' 


•113559 


•073935 


i 11 


•126936 


•1035S5 


12 


•110584 


•043G9.3 


12 


•113696 


'074487 


! 12 


-12707S 


•104070 


13 


•110717 


•044218 


13 


•118834 


•074941 


i 13 


127-2-20 


■104555 


14, 


•110850 


•044735 


14 


•118972 


•075453 


i 14 


•127362 


•105040 


15 


•110983 


•045258 


15 


•119110 


•075946 


i 15 


•127504 


105523 


16 


•111116 


•045777 


16 


•119247 


•076448 


i 16 


12764G 


•106008 


ir 


•111249 


•046297 ^ 


17 


•119385 


•07G948 


' 17 


-127TS9 


-106491 


18 


•111383 


•046818 


18 


•119523 


•077450 


i 18 


•127931 


•106974 


19 


•111516 


•047338 , 


19 


•119661 


•077951 


! 19 


•12S073 


•107457 


20 


•111650 


•047857 ^ 


20 


•119799 


•078452 


1 20 


•128216 


•107940 


21 


•111783 


•048377 


21 


•119987 


•078952 


i 21 


•128858 


•108423 


22 


•111917 


•048396 


22 


•120075 


•079452 


j 22 


•12S5ul 


■1CS906 


23 


•112051 


•049415 


23 


•120213 


•079951 


28 


•128643 


•109387 


24 


•112185 


•049933 


24 


•120351 


•080451 


1 24 


•128786 


•109869 


25 


•112319 


•050451 


25 


•120490 


•08r951 


1 25 


•128929 


•110350 


26 


•112453 


•050968 


26 


•120628 


•081449 


' 26 


•129072 


•110831 


2T 


•112587 


•051487 


27 


•120767 


•081947 


27 


•129215 


•111312 


28 


•112721 


•052004 


28 


•120905 


•082445 


1 28 


•129358 


•111793 


29 


•112855 


•052520 


29 


•121044 


•082943 


1 29 


•129501 


•112273 


30 


•112989 


•C53037 


30 


•121183 


•083441 


1 30 


•129644 


•112754 


31 


•113124 


•058553 


31 


•121322 


•083938 


i 31 


•129788 


•113233 


32 


•113258 


•054069 


32 


•121461 


•(84485 


I 82 


•129931 


•113712 


33 


•118393 


•054584 


33 


•121600 


•084982 


i 33 


•130074 


•114192 


34 


•113527 


•C55C99 


34 


•121789 


•085428 


; 34 


•130218 


•114671 


35 


•113662 


•055C14 , 


35 


•121878 


•C85924 


1 35 


•130362 


•115149 


36 


•113797 


'056129 


36 


••122017 


•086420 


1 30 


•180505 


•115627 


37 


•113931 


•056642 : 


37 


•122157 


•086916 


37 


•130649 


•116105 


38 


•114066 


•057157 


38 


•122296 


•C87411 


1 38 


•130793 


•116583 


39 


•114201 


•057670 


39 


•122435 


•0879C5 


89 


•130936 


•117060 


40 


•114336 


-053183 i 


40 


•12-2575 


•088401 


40 


•131080 


•117537 


41 


-114471 


•058696 ^ 


41 


•122714 


•083895 


! 41 


•1312-24 


•118015 


42 


•114606 


-059208 ■ 


42 


•122854 


•C89388 


I 42 


•131368 


•118491 


43 


•114742 


'059721 ^ 


43 


•122994 


•089882 


! 48 


•131513 


•118968 


44 


♦114877 


-060282 


44 , 


'128134 


•C90376 


i 44 


•131657 


•119443 


45 


-115013 


-060745 1 


45 


•123273 


•090869 


i 45 


•181801 


•119919 


46 


•115148 


-061256 i 


46 


•123413 


•091361 


! 46 


•131945 


•120394 i 


47 


•115288 


•061766 i 


47 


•123553 


•091854 


I 47 


•182C90 


•120870 


48 


•115419 


'062277 


48 


'123698 


•C92346 


43 


•132235 


•121345 


49 


•115555 


-062788 


49 


•123834 


'092888 


49 


•132379 


•121819 


50 


•115691 


•063398 


. 50 


-123974 


•093329 


! 50 


•132524 


•122294 ! 


51 


•115827 


-063807 


' 51 


•124114 


•C93S21 


51 


•132669 


•122768 1 


52 


'115963 


•064316 


5^ 


•124255 


•(9J^12 


52 


•132814 


•123242 ' 


53 


•11609S 


•064826 


53 


•124395 


•094802 


53 


■182958 


•123715 


54 


•116235 


-065335 


54 


•124535 


•095293 


! 54 


•183103 


•124188 


55 


•116871 


-065843 


55 


-124676 


•095783 


s 55 


•13824S 


•1-24661 


56 


•116507 


-066351 


56 


•124817 


•096272 


: 56 


•133394 


•1-25135 


57 


'116643 


•066859 


57 


•124958 


•096763 


! 57 


•138539 


•125607 


58 


-116779 


-067366 


58 


•125098 


•097251 


i 58 


•183684 


•126079 


59 


•116916 


•067874 


59 


•125239 


•097740 


59 


433830 


•126551 


60 


'117052 


-068380 


60 


•125380 


•098229 


60 


•133975 


•127022 



External Secants. 



435 



27 DEGEEES. 



Nn!. .No 



irithm. 



28 DEGEEES. 



29 DEGEEES. 





1 

2 

8 

4 

6 

G 

7 

8 

9 

10 

11 

12 

13 

U 

15 

16 

17 

IS 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

■29 

30 

81 

82 

83 

34 

85 

86 

87 



40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
CO 



0-122327 
•122493 
•122060 
•122826 
•122993 
•123160 
•123327 
•123495 
•123662 
•123829 
•128997 
•124165 
•124333 
•124501 
•124669 
•124838 
•1250(:6 
•125175 
•12.5344 
•125513 
•125682 
•125851 
■126021 
•126191 
•126360 
•126530 
•126700 
•120871 
•127041 
•127211 
•127382 
•127558 
•127724 
•127895 
•128066 
•12S237 
•J284f9 
•128581 
•128753 
•128925 
•1291,97 
•129269 
•129441 
•129614 
•129787 
•129960 
•130132 
•130305 
•130479 
•130652 
•13f,826 
•13(999 
•131173 
•131347 
•131522 
•131696 
•131871 
•132046 
■132220 
•132395 
•182570 



9(87520 
•C83111 
•(■88700 
•( 89290 
•C89S79 
•(:904G9 
•081(57 
•(91647 
•(92236 
•(92823 
•C 93410 
•C9399S 
•(94585 
•C95173 
•C95760 
•C96346 
•C9G932 
•097579 
•(98103 
•C9SC88 
•(99273 
•(99858 
•100442 
•101027 
-101610 
•102194 
•102776 
•103361 
•10S944 
•104525 
•105108 
•105C90 
•100272 
•106853 
•107434 
•108014 
•1(8596 
•1(9175 
•1(9756 
•110.335 
•11(914 
•111493 
•112072 
•112651 
•113228 
•118808 
•114385 
•114962 
•115530 
•116117 
•116694 
•117269 
•117845 
•11 8422 
•118998 
•119573 
•120148 
•12(»723 
•121297 
•121872 
•122446 



Nat No. I Lngari.lim. j Min. | Nat No 



0^132570 
•132745 
•13-2920 
•133C96 
•133272 
•133448 
•133624 
•133800 
•1SS976 
•134153 
•134330 
•134506 
•134683 
•1348G0 
•185(37 
•185215 
•135C92 
•135570 
•135748 
•135926 
•136104 
•1362R2 
•136460 
•136639 
•136S1S 
•13C997 
•137176 
•137355 
•137534 
•137713 
•137893 
•138073 
'138253 
•138433 
•138613 
•138793 
•138974 
•189155 
•139386 
•139517 
•139698 
•139880 
•140061 
•140242 
•140424 
•140606 
•140788 
•14C970 
•141153 
•141.336 
•141518 
•141701 
•141884 
•142067 
•142250 
•142434 
•142{)1S 
•142802 
•142986 
•143170 
•143354 



•122445 





•123019 


1 


•123593 


2 


•124106 


3 


•124733 


4 


■125312 


5 


•125884 


6 


•126453 


7 


•127028 


8 


•127600 


9 


•128171 


10 


•128742 


11 


•129312 


12 


•129888 


13 


•180453 


14 


•131024 


15 


-181.C94 


16 


•132162 " 


17 


•132732 


IS 


•133301 


19 


•133870 


20 


•1.344r8 


21 


•1350(.6 


22 


•135574 


28 


•130142 


24 


•136710 


25 


•137277 


26 


•187843 


27 


1.384(9 


28 


•138976 


29 


•139542 


80 


•1401(8 


81 


•14(674 


82 


•141239 


83 


•141804 


34 


•142369 


85 


•142934 


86 


•143490 


37 


•144068 


38 


•144626 


89 


•145191 


40 


•145754 


41 


•146316 


42 


•146879 


43 


•147442 


44 


•148005 


45 


•148566 


46 


•149128 


47 


•149690 


48 


•15(251 


49 


•15r813 


50 


•151378 


51 


•151984 


52 


•152494 


58 


•158054 


54 


•153614 


55 


•154178 


56 


•154734 


57 


•155292 


58 


•155851 


1 59 


•156410 


CO 



0-143354 
•143538 
•143723 
•143908 
•144(93 
•144278 
•144463 
•144649 
•144834 
•145020 
•145206 
•145391 
•145578 
•145764 
•1459.50 
•146187 
•146324 
•146511 
•146C98 
•146685 
•147072 
•147260 
•147448 
•147636 
•147825 
•148; 12 
•148200 
•148389 
•148577 
•148766 
•14S956 
•149145 
•149334 
•149524 
•149714 
•140908 
•150( 93 
•150283 
•150474 
•150664 
•150854 
•151045 
•151236 
•151427 
•151619 
•151810 
•152001 
•152198 
•152385 
•152577 
•152769 
•152902 
•153154 
•153347 
•153540 
•153738 
•158920 
•154120 
•154818 
•154507 
•154700 



T.ogai 



9^156410 
•156968 
•157527 
•15SC84 
•158642 
•159199 
•159757 
•160314 
•160869 
•161427 
•161983 
•162539 
•163(95 
•163650 
•164206 
•164760 
•165315 
•165869 
•166423 
•166977 
•167531 
•168085 
•1G86S9 
•169191 
•109749 
•170296 
•170840 
•171401 
•171953 
•172505 
•173(57 
•173608 
•174158 
•174710 
•175261 
•175810 
•176360 
•176910 
•177460 
•178008 
•178557 
•179107 
•179655 
•180205 
•180752 
•181300 
•181847 
•182395 
1S2948 
•18.3489 
•184036 
•184588 
•185129 
•185675 
•18()221 
•186766 
•187318 
•187858 
•188408 
•18894T 
•189491 



43G 








Versed 


Sines. 








30 DEGREES. 


j 81 DEGREES. [ 


8 


2 DEGREES. 


Mill 


Nat. No. 


LoRaiiMini. 


' Mill. 


Nat No 


,..:...,■.. i 


^Vm. 


Na( No 


I...p»nthm 


' 


0-133975 


9-127(22 





0-142833 


9^154S27 ! 





0-151952 


9^1 81706 


1 


•134120 


-127494 


1 


•142983 


•155283 i 


1 


•152106 


-182147 


2 


•134266 


-127964 


2 


•143133 


•155788 1 


2 


•152261 


•182587 


8 


•134412 


-128436 


3 


•143282 


-156192 ; 


3 


•152415 


•183027 


4 


•134558 


■128906 , 


4 


•143482 


•156G47 ; 


4 


•152569 


183466 


5 


•134703 


•129876 1 


5 


•143583 


■1571(2 


6 


•152724 


•183906 


6 


•134849 


-129846 j 


6 


•143733 


•157556 


6 


•152878 


•184845 


T 


•134995 


-130316 


7 


•148883 


-158C10 


7 


•153033 


184784 


8 


•185141 


•1307S6 ! 


8 


•144034 


•158464 


8 


•158188 


185223 


9 


■185287 


•131254 1 


9 


•144184 


-158917 


9 


•158842 


-185661 


10 


•135483 


-1817-24 ; 


10 


•144335 


-159370 


10 


•158497 


•186100 


11 


•135579 


•132192 ! 


11 


•144485 


•159823 


11 


•158652 


•186538 


12 


•135725 


•132660 ! 


12 


•144686 


•16G275 : 


12 


•153807 


•186975 


13 


•185872 


•183128 1 


13 


•144787 


•160728 


13 


•158962 


•187413 


14 


•136018 


•138596 


14 


•144937 


■161180 : 


14 


•154117 


-187850 


15 


•136165 


•184064 


15 


•145088 


-161G82 i 


15 


•154272 


-188287 


, 16 


•136811 


•184531 


16 


•145289 


•162C83 ; 


16 


•154427 


■188724 


! IT 


•136458 


•184998 


17 


•145890 


•162535 


17 


•154583 


189161 


i ^^ 


•136605 


•185465 


18 


•145541 


•162986 : 


18 


•154788 


189598 


\ 19 


•136751 


185981 


19 


•145692 


•163486 ; 


19 


•154S94 


190033 


; 20 


•136S9S 


•136397 


20 


•145844 


•168887 ' 


20 


•155G49 


•190469 


\ 21 


•137045 


•136863 ' 


21 


•145995 


-164837 : 


21 


■155205 


•19G905 


22 


•137192 


-187829 


22 


•146146 


-1647SS ■ 


22 


•155360 


191340 


23 


•137839 


-137794 


23 


•146298 


•165288 


23 


•155516 


•191775 


24 


•137486 


-138259 


24 


•14G450 


•1G5687 


24 


•155672 


19-2210 


25 


•137683 


-188724 


25 


-146601 


•166187 ; 


25 


•155828 


192045 


26 


•137781 


•185189 


26 


•146752 


•166585 


26 


•155984 


■193080 


27 


•137928 


•139658 


27 


•14C9C4 


167034 : 


2T 


•156140 


•198514 


28 


•188076 


-140117 


28 


•147056 


-167483 


28 


•156296 


•193948 


1 29 


•188228 


-14G5S1 


29 


1472(8 


-167981 i 


29 


•156452 


-194382 


; 30 


•138371 


•141045 


30 


•147860 


-168879 


80 


•1566(9 


•194815 


81 


•138518 


•141507 


81 


-147512 


168827 ■ 


81 


•156765 


195249 


32 


•138666 


141971 


82 


•147664 


-1G9274 ^ 


82 


•156921 


195682 


83 


•188814 


142434 


83 


•147817 


•1G9722 


83 


•1570TS 


-196115 


34 


•188962 


•142890 


34 


•1479C9 


170169 


84 


•157234 


-196547 


; S5 


•139110 


-148858 


35 


•148121 


■170015 


85 


•157391 


-196979 


36 


•189258 


•148820 


86 


•14S273 


171068 


86 


•157548 


•197412 


! 37 


•1394C6 


-144282 


. 37 


•158426 


-1715C9 


37 


•1577(5 


-197&44 


* 38 


•139554 


•144T48 


; 38 


•148578 


171954 


38 


•157S61 


•198275 


39 


•189703 


-145-204 


; 89 


•148730 


■172400 


89 


•158018 


•198707 


40 


•139851 


•145665 


' 40 


•148S83 


-172846 


40 


•158175 


•199138 


41 


•189999 


-146125 


1 41 


•149036 


•178292 


41 


•158832 


•199569 


42 


•140147 


-146585 


' 42 


•149189 


-173736 


42 


•158490 


-200000 


43 


•14G296 


-147046 


43 


•149842 


•174181 


43 


•158647 


•200480 


44 


•14G445 


•147506 


; 44 


•149495 


•174626 


44 


•158804 


■200360 


45 


•140594 


-147965 


; 45 


•149648 


•175070 


45 


•158962 


•201290 


46 


•140742 


•148424 


; 46 


•1498G1 


•175514 


46 


•159119 


-201720 


47 


•140891 


•148883 


' 47 


•149954 


•175958 


47 


•159276 


2G2149 


48 


•141040 


-149842 


i 48 


•150107 


•176401 


48 


•159483 


202579 


49 


•141189 


•149801 


j 49 


•150261 


•176S45 


49 


•159591 


•203008 


50 


•141838 


•15C259 


' 50 


•150414 


•177288 


50 


•159749 


•208487 


51 


•14148T 


-1507C7 


1 51 


•150568 


-177731 


51 


•159907 


203866 


52 


•141686 


-151175 


! 52 


•150721 


•178174 ; 


52 


•160065 


-2G4295 


53 


•141786 


•151632 


53 


•150875 


•178616 


53 


•16C223 


•204723 


54 


•141935 


•152089 


1 54 


•151028 


•179058 ■ 


54 


•16C380 


•2C5151 


55 


•142085 


•152546 


' 55 


•151182 


•179500 ! 


55 


•160538 


•205579 


56 


•142284 


■153003 


' 56 


•151336 


•179942 ; 


56 


•160697 


•2G6006 


57 


•142884 


•153460 


i 57 


-151490 


•180884 


57 


•160855 


•2C6483 


58 


•142583 


•158916 


1 58 


•151644 


•180824 


58 


•161013 


■206860 


59 


•142683 


-154372 


59 


•151798 


•181265 


69 


•161171 


•207287 


60 


•142838 


•154827 


60 


•151952 


•181706 


60 


•161330 


•207714 









E.VTPTIVAT, ^ 


Decants. 






437 














80 DEGREES. 


iMi:i. 


31 DEGREES. 




32 DEGREES. 


Min. 


i Nat. No. 


1 Lo-ari;iim 


1 

1 NHt. No 


Loearitlim. 


Min. 


i Nat. No. 


! L„,.ui..... 





. 0-154700 


i 9-130491 





1 0-163638 


9-221761 





0-179179 


1 9-253286 


1 


1 -154894 


! •190086 


1 


' -166337 


•222293 


1 


-179898 


•253805 


2 


, -155089 


i -190579 


2 


: -167041 


•222824 


2 


•179608 


•254824 


3 


! -155283 


! -191124 


3 


i -167245 


•223854 


8 


•179322 


•254844 


4 


•15547S 


i -191668 


4 


•167450 


•228885 


4 


•180037 


•255362 


5 


•155i37o 


1 -192211 


5 


•167655 


•224417 


5 


-180252 


•255881 


6 


•155867 


•192754 


6 


•167360 


•224947 


1 6 


•180467 


•256399 


7 


•156062 


-193297 


7 


•163065 


•225477 


■7 


•180683 


•256917 


8 


•156257 


•193840 


8 


•163270 


•226007 


1 8 


•180899 


•257436 


9 


•153452 


•195382 


9 


•16S476 


•226587 


! 9 


•181115 


•257953 


1) 


•156G48 


•194925 


10 


•168681 


•227066 


10 


•181331 


•258471 


11 


•15G344 


-195467 


11 


•163387 


•227595 


11 


•181547 


•253989 


12 


•157040 


-196008 


12 


•169098 


•228124 


12 


•181763 


•259505 


13 


•1572-38 


-196550 


13 


•169299 


•228653 


18 


•182080 


•260023 


14 


•157482 


-197*91 


14 


•169505 


•229182 


14 


•132197 


•260540 


15 


•15763 S 


•197633 


15 


•169711 


•229711 


15 


•182413 


•261056 


16 


•157324 


-193174 


16 


•169918 


•230238 


hi6 


•182680 


' •261573 


IT 


•15SJ21 


•193714 


17 


•170125 


•230767 


1 17 


•182348 


•262390 


18 


•158218 


•199255 


18 


■170382 


•231295 


18 


•183066 


•262607 


19 


•153115 


•199795 


19 


•170539 


-231822 


19 


•183283 


•268122 


2i) 


•153312 


•200385 


20 


•170746 


-282350 


20 


•183501 


•283638 


21 


•1538J9 


•200S75 


21 


•170958 


-282377 ; 


' 21 


•188719 


•264154 


22 


•159 ;07 


•201415 


22 


•171161 


•283405 ' 


22 


•188937 


•264669 


23 


•159204 


•201914 


23 


•171369 


•283982 


23 


•184156 


-285184 


24 


-150102 


•232498 


24 


•171577 


•284458 


24 


•184374 


•265699 


25 


•159600 


•203032 . 


25 


•171785 


•234935 , 


25 


•184593 


•266214 


26 


•159798 


•203571 


26 


•171993 


•285510 , 


26 


•184312 


•286729 


27 


•150997 


•204110 


27 


•17-2201 


•236036 , 


27 


•185031 


•267244 


28 


•160195 


•204648 


28 


•172410 


•286562 


23 


•185250 


•267758 


29 


•160398 


•205186 


29 


•172619 


•237088 ; 


29 


•185469 


•268272 


30 


•160592 


•205725 


80 


•172S28 


•237613 


80 


•1S5689 


•268786 


81 


•16)791 


-206261 


31 


•173037 


•288139 i 


81 


•185909 


•269800 


32 


•16)901 


•206800 


82 


•173246 


•238663 i 


82 


•188129 


•269814 


83 


•161190 


•207387 


83 


•178456 


•289189 ] 


83 


•186349 


•270327 


84 


•161390 


•207874 


84 


•178665 


•289718 1 


34 


•186570 


•270840 


85 


•161589 


•203410 


35 


•173375 


•240287 ! 


35 


•186790 


•271358 


86 


•161789 


•208947 


36 


•174035 


•240763 1 


86 


•187011 


•271867 


87 


•161939 


•2;9484 


87 


•174295 


•241288 1 


87 


•187232 


•272379 


33 


•162189 


•210020 


38 


•174505 


•241809 i 


88 


•187453 


•272391 


39 


•162389 


•210555 


89 


•174716 


•242333 1 


89 


•187674 


•273404 


40 


•162589 


-211091 


40 


•174927 


•242357 ' 


40 


•187898 


•273916 


41 


•162789 


•211626 


41 


•175038 


•243381 ! 


41 


•183117 


•274423 


42 


•162990 


•212161 


42 


•175349 


•248903 ; 


42 


•138839 


•274940 


43 


•163191 


-212697 


43 


•175560 


•244426 


43 


•183561 


•275451 


44 


•163892 


•213282 j 


44 


•175772 


•244949 


44 


•188783 


•275983 


45 


•168594 


•218766 


45 


•175988 


•245471 1 


45 


•189006 


•276475 


46 


•163795 


•214801 1 


46 


•176195 


•245993 i 


46 


•189228 


•276985 


47 


•163997 


•214885 1 


47 


•176407 


•246516 


47 


•139450 


-277495 


48 


•164198 


•215868 


48 


•176619 


•247037 i 


48 


•189673 


•278006 


49 


•164400 


•215903 1 


49 


•176882 


•247559 ! 


49 


•139396 


•278517 


50 


•164603 


•216437 


60 


•177044 


•248081 i 


50 


•190120 


•27902S 


51 


•164805 


•216970 


51 


•177257 


•248602 i 


51 


•190344 


•279588 


52 


•165008 


•217504 


52 


•177570 


•219124 ' 


52 


•190568 


•230049 


53 


•165210 


•218036 I 


58 


•177233 


•249844 ' 


53 


•190792 


•280559 


54 


•165413 


•218569 ! 


54 


•177896 , 


•250165 , 


54 


•191015 


•231 068 


55 


•165616 


•2l0i()l 1 


55 


•1781(9 


•250685 ; 


55 


•191240 


•281578 


56 


•165S19 


•219684 1 


5() 


•178822 


■251206 1 


56 


•191464 


•2^2087 


57 


•166028 


•220167 


57 


•178586 


•251727 : 


57 


•19 1 689 


•232596 


58 


•166226 


-220699 


58 


•17>^750 


•252246 1 


58 


•191914 


•2sni05 


59 


•1664.80 


•221281 


59 


•178964 , 


•252766 ! 


59 


•1921S9 


•283614 


60 


•166633 


•221761 j 


60 


•179179 , 


•258286 1 


60 


•192864 


•284123 



438 








Versed 


Sines. 








1 

83 DEGREES. 




34 DEGREES. 


35 DEGREES. 


Min. 


Nat. No 


{ Logarithm. 


1 .in. 


1 Nat. Nc. 


1 Loicarithm. 


Min. 


1 Nat. No. 


! Logarithm. 





0-161330 


i 9-207714 





0-17G963 


9-232901 





0-180848 


■ 9-257313 


1 1 


•161488 


•208140 


1 


•171126 


! •283314 


1 


•181015 


•257714 


2 


•161647 


-208566 


2 


•171289 


j -283727 


2 


•181182 


-258114 


3 


•161805 


•238992 


3 


•171452 


-234189 


3 


•181349 


-258514 


i 4 


•161984 


1 -209418 


4 


•171614 


•234552 


4 


•181516 


•258914 


5 


•162123 


-209843 


5 j ^171777 


i -234964 


5 


•181683 


•259314 


6 


•1622S2 


•210269 


6 


1 ^171940 


j -235876 


6 


•181850 


•259714 


T 


•162440 


-210693 


7 


•172103 


-285788 


7 


•182018 


-260114 


8 


•162599 


-211118 


8 


•172266 


•236199 


8 


•182185 


-260513 


9 


•162758 


•211543 


9 


•172480 


•236611 


9 


•182353 


•260912 


10 


•162917 


•211967 


10 


1 -172593 


•237022 


10 


•182520 


•261810 


i 11 


•163076 


•212391 


11 


-172756 


-237433 


11 


•182688 


1 -261769 


; 12 


•163285 


•212814 


12 


1 -172920 


•287844 


12 


•182855 


1 -262107 


1 13 


•163395 


•213238 


18 


: -173088 


•288255 


13 


•183023 


i -262505 


i 14 


•163555 


-218662 


14 


•178247 


-288665 


14 


•183191 


i -262903 


! 15 


•163714 


-214085 - 


15 


•178410 


-289075 


15 


•183859 


i -263301 


^ 16 


•163874 


-214509 i 


16 


•178574 


•239485 


16 


•188527 


1 -263699 


; IT 


•164033 


•214931 


17 


•173788 


•289895 


17 


•183695 


•264096 


: 18 


•164193 


•215353 


18 


•178932 


•240304 


18 


•188863 


-264498 


; 19 


•164353 


•215776 


19 


•174066 


•240T13 


19 


•184031 


•264890 


' 20 


•164512 


•216198 


20 


•174230 


•241122 i 


20 


•184199 


•265286 


j 21 


•164672 


•216620 


21 


•174394 


•241531 1 


21 


•184367 


•265683 


i 22 


•164832 


•217042 


22 


•174558 


-241940 ; 


22 


•184535 


•266080 


! 23 


•164992 


•217462 


28 


•174723 


•242849 


23 


•184704 


•266476 


: 24 


•165152 


•2178S4 


24 


•174887 


•242757 ' 


24 


•184872 


•266871 


' 25 


•165312 


•218305 


25 


•175051 


•243165 i 


25 


•185041 


-267267 


23 


•165473 


-218726 


26 


•175216 


-243572 ' 


26 


•185210 


•267663 


27 


•165683 


•219146 


27 


•175880 


•243980 


27 


•185878 


•268058 


28 


•165793 


-219567 


28 


•175544 


•244387 I 


28 


•185547 


•268453 


29 


•165954 


•219987 ; 


29 


-1757C9 


•244794 i 


29 


•185716 


-268848 


30 1 


•166114 


-220406 


80 


•175874 


•245201 ! 


30 


•185885 


•269243 


31 1 


•166275 


-221S26 : 


31 


•176039 


•245608 i 


31 


•186054 


-269637 


32 1 


•166436 


-221246 


32 


•176234 


•246014 1 


32 


•186223 


-270032 


83 1 


•166597 


•221665 i 


33 


•176869 


•246421 \ 


33 


•186392 


•270426 


34 ! 


•1G6757 


•222084 ' 


84 


•176584 


•246827 ' 


34 


•186561 


-270820 


85 


•166918 


•222502 


35 


•176699 


•247233 ! 


35 


•186730 


•271214 


36 


•167079 


•222922 i 


36 


•176864 


-247689 , 


36 


•186900 


-271608 


sr 


•167240 


-228340 i 


87 


•177029 


•248044 1 


87 


•187069 


-272001 


88 


•167401 


•223758 1 


38 


•177194 


•248449 


38 


•187288 


•272394 


89 i 


•167562 


•224175 1 


89 


•177360 


-248S55 


39 


•187408 


•27278T 


40 


•167723 


-224593 1 


40 


•177525 


•249259 1 


40 


•187577 


-273180 


41 


•167885 


-225011 


41 


•177690 


•249664 i 


41 


•187747 


•273573 


42 


•168046 


•225427 ! 


42 


•177856 


•250069 i 


42 


•187917 


•278965 


43 


•168208 


-225845 ; 


43 


-178022 


•250473 1 


48 


•188087 


•274857 


44 


•168369 


•223262 1 


44 


•178187 


-250876 ! 


44 


•188256 


•274749 


45 


•168530 i 


•226678 j 


45 


•178353 


•251280 ^ 


45 


•188426 


•275140 


46 


•168692 i 


•227094 


46 


•178519 


-251684 ^ 


46 


•188596 


-275532 


47 


•168354 


•227511 1 


47 


•178685 


-252087 ' 


47 


•188766 


-275924 


48 


•169016 


-227927 1 


48 


•178851 


•252491 1 


48 


•188936 


•276315 


49 


•169178 


•22S342 i 


49 


•179017 


•252894 


49 


•189106 


-276706 


50 


•169340 i 


-228758 i 


50 


•179188 


•258297 


50 


•189277 


-277C97 


61 


•1695G2 


•229173 ! 


51 


•179849 


•258699 


61 


•189447 


•277487 


52 


•169663 


•229587 ! 


52 


•179515 


•254101 1 


52 


•189617 


•277878 


53 


•169S26 


•230003 j 


53 


•179682 


•254504 i 


53 


•189788 


-278268 


64 


•169983 


•230418 1 


64 


•179S48 i 


-854906 ; 


54 


•18995S 


•27865S 


55 


•170150 


•230832 i 


55 


•180015 i 


•255808 


55 


•190129 


•279048 


56 


•170812 


•231246 


56 


•180182 I 


-255710 ! 


56 


•190300 


-27943S 


67 


•170475 


-281660 


57 


•180348 


•256111 i 


57 


•190471 


•279828 


58 


•170638 


•282074 


58 


•180515 


•256512 


58 


•190641 


•280217 


59 


•170800 


•232487 


59 


-180681 


•256913 


59 


•190812 


•280606 


60 


•170963 


•282901 


60 


•180848 


•257313 


60 


•190983 


•280995 

1 









E. 


v:tekxal 


b EC A NTS. 






439 


83 DEGREES. 


1 34 DEGEEES. 


i Min. 


35 DEGEEES. 


Min. 


I Naf. No, 


1 Logarillim. 


i 


1 Nat. No. 


1 J.ogaiid.m. 


1 Nat No. 


Logari;h n. 


r 


0-192364 


9-284123 


i 


0-206218 


9 ■314326 





0-220775 


9-843949 


i 1 


•192589 


•284631 


I 1 


•206455 


•314S25 


1 


•221024 


•344488 


! 2 


•192814 


•285189 


i 2 


•206692 


•315323 


2 


•221272 


•844927 


i 3 


•193040 


■285647 


1 ^ 


•20C929 


•315821 


3 


•221521 


•345415 


4 


•193266 


•286155 


4 


•207167 


•816319 


4 


•221771 


•845904 


5 


•193492 


•286662 


5 


•207404 


•816817 


5 


•222C20 


•346892 


6 


•193718 


•287171 


1 6 


•207642 


•317315 


6 


•222270 


•346881 


T 


•193945 


•287677 


7 


•207880 


•317812 


7 


•222520 


-847370 


1 8 


•194172 


•238185 


8 


•208118 


•8183(9 


8 


•222771 


•847858 


9 


•194399 


•288692 


9 


•208356 


•818806 


9 


•223C21 


•348346 


10 


•194G25 


•289199 


10 


•206594 


•3193C2 


10 


•223272 


•348888 


11 


•194S52 


•289705 


11 


•208S33 


•319799 


11 


•223523 


•849321 


12 


•195080 


•290211 


12 


•2C9072 


•320296 


12 


•223774 


•349808 


13 


•195307 


•290718 


13 


•2C9311 


•320793 


13 


•224025 


•850295 


14 


•195535 


•290224 


14 


•209550 


•321288 


14 


•224276 


•350782 


15 


•195763 


•291730 


15 


•2G9790 


•821785 


15 


•224528 


•351270 


IG 


•195992 


•292237 


16 


•210C30 


•322280 


^16 


•225780 


•851757 


IT 


•196220 


•292742 


17 


•210270 


•322777 


17 


•225031 


•852243 


18 


•196448 


•293247 


18 


•21C510 


•823272 


! 18 


•225284 


•852730 


19 


•196677 


•293753 


19 


•210750 


•323767 


19 


•225536 


•853216 


20 


•l9G9a6 


•294258 


20 


•210991 


•824263 


20 


•225789 


•358702 


21 


•197135 


•294763 


21 


•211231 


•324758 


21 


•226042 


•854189 


22 


•197364 


•295268 


1 22 


•211472 


•825253 


1 22 


•226295 


•354675 


23 


•197593 


•295771 


j 23 


•211714 


•325749 


23 


•226548 


•355161 


24 


•197823 


•29G277 


; 24 


•211955 


•326243 


24 


•226801 


-355646 


25 


•198053 


•296781 


25 


•212197 


•326738 


25 


•227C55 


-856131 


26 


•198283 


•297285 


26 


•212438 


•327282 


26 


•227310 


•856617 


2T 


•198513 


•297789 


27 


•212680 


•327726 


27 


•227564 


•357102 


28 


•198744 


•298293 


28 


•212922 


•328220 


28 


•227818 


•857587 


29 


•198974 


•298797 


29 


•213164 


•828713 


29 


•228072 


•358072 


30 


•199205 


•299299 


30 


•213457 


•329207 


80 


•228826 


•858557 


31 


•199136 


•299803 ! 


31 


•213650 


•829701 


81 


•228581 


•359041 


32 


•199667 


•300307 


32 


•213892 


•830194 


82 


•228837 


•359526 


33 


•199899 


•3008C9 


33 


•214135 


•330688 


33 


•229C92 


•360011 


34 


•200130 


•301312 


34 


•214^79 


•331181 


84 


•229848 


•860495 


85 


•200362 


•301815 


35 


•214622 


•831674 


35 


•229604 


•36C979 


36 


•20C594 


•3C2317 


36 


•214866 


•332167 


36 


•229860 


•361468 


37 


•200826 


•302820 


37 


•215110 


•882G59 ! 


87 


•230116 


•361947 


88 


•201058 


•303322 


38 


•215354 


•333152 1 


88 


•23C373 


•362431 


89 


•201290 


•303823 ; 


89 


•215598 


•833644 


89 


•230630 


•362914 


40 


•201523 


•304325 i 


40 


•215842 


•834136 


40 


•230886 


•363398 


41 


•201756 


•304827 


41 


•216(!87 


•334629 


41 


•231148 


•363881 


42 


•201990 


•305328 


42 


•216332 


•885121 


42 


•231400 


•864364 


43 


•202223 


•305830 


43 


•216577 


•835613 


43 


•231658 


•364847 


44 


•202456 


•300331 


44 


•216822 


•836104 i 


44 


•231916 


•365330 


45 


•202690 


•306832 


45 


•217068 


•336r.95 i 


45 


•282173 


•365812 


46 


•202924 


•307333 


46 


•217313 


•387086 1 


46 


•232431 


•366295 


47 


•203158 


•307834 


47 


•217559 


•337577 ! 


47 


•232690 


•366778 


48 


•203392 


•308334 


48 


•217806 


•338069 


48 


•232949 


•367260 


49 


•203626 


•308834 


49 


•218052 


•838560 


49 


•233207 


•867742 


50 


•203861 


•309334 


50 


•218299 


•339050 


50 


•233466 


•368224 


51 


•204(196 


•309334 


51 


•218545 


•839541 1 


61 


•233726 


•868706 


52 


•204331 


•310334 


52 


•218792 


•340031 j 


62 


•233985 


•369188 


58 


•204568 


•310834 


53 


•219040 


•340522 i 


53 


•234245 


•869670 


54 


•204801 


•311333 


54 


•219287 


•341012 


54 


•234505 


•370151 


65 


•205037 


•311832 


55 


•219535 


•841502 


55 


•234764 


•870632 


66 


•205273 


•312381 


56 


•219782 


•841992 


56 


•235025 


•871114 


67 


•205509 


•312830 


57 


•220030 


•842481 


57 


•235285 


•371595 


58 


•2;)5745 


•813329 


58 


•220278 


•342970 


58 


•235546 


•372076 


59 


•205981 


•813823 


59 


•220526 


•848460 


59 


•235807 


•372567 


60 


•206218 


•314326 


60 


•220775 


•8-48949 


60 


•286068 


•873087 



440 






Versed Sixes. 








36 DEGREES. | 


37 DEGEEEZ. 


33 DEGREES. 


Min. 


Nat No 


Lo..,=-.hm 1 


M.. 


Nat. No. 


Lo-aritiim. 


Min. 


N:.t No 


Lo-a-ihrt.. 





0-19 953 


9-2S:095 





0-201365 


9-303933 





0-211990 


9-326314 


1 


•191154 


•2S1384 


1 


•231540 


•304361 


; 1 


-21 21 09 


•326C81 


2 


•191325 


•281772 


2 


•201715 


•304737 


! 1 


•21-2:348 


•327047 


3 


•191496 


•232160 


3 


•201890 


•305115 


•212527 


•327414 


4 


•191G67 


•2S2548 


4 


•202065 


•305492 


i 4 


•212706 


•327780 


5 


•191839 


•2S2936 


5 


•23-2240 


•305868 


! 5 


•212886 


•323146 


6 


•192010 


•2333-24 


6 


•202416 


•300245 


1 6 


•213065 


-323512 


T 


•1921S1 


•2S3711 


7 


•2:2591 


•306621 


7 


•21:3244 


-325877 


8 


•192353 


•234G99 


8 


•202767 


•306998 


8 


•21:34-24 


•329243 


9 


-192525 


-284486 


9 


•202943 


•307874 


9 


•21:3803 


•3296C8 


10 


•192796 


•254873 


10 


•2 03118 1 


•307749 


10 ! 


•213783 


•329974 


11 t 


•192808 


•285260 


11 


•2;3-294 


•308125 


11 1 


•213963 


•33CSS9 


12 


•193040 . 


•285947 


12 


•2)3470 


•308501 


12 . 


•214143 


•330704 , 


18 


•193212 


•283C33 


13 


•233646 


•308376 


13 


•214323 


-331G69 


; 14 


•19:3334 1 


•23^1419 


14 


•2.^3322 


•3C9251 


14 


-214503 


-3:314:33 


i 15 


•193555 ! 


•28'3805 


15 


•20399S 


•309625 


15 


-214633 


•;3:31798 


i 18 


•193727 ! 


•2S7191 


16 


•204174 


•310000 


16 


•214368 


-3:32162 


1 17 


•198900 


•237576 


17 


•204350 


•310375 


17 


-215i348 


-332526 


13 


•191072 


•287982 


13 


-234527 


•310749 


18 


•215224 


•332890 


: 19 


•191244 ' 


•2S3347 


19 


•204708 


•311124 


19 


•215404 


•383254 


, 20 


•194416 1 


•2S8732 


20 


•204830 


•311498 


20 


■215585 


•333618 


; 21 


•194583 ' 


•239117 


21 


•235056 


•311571 


21 


•215765 


•333980 


; 22 


•194761 


•230502 


90 


•2,;5237 


•312248 


22 


•215945 


•334344 


i 23 


•194934 


•2S9387 


23 


•235409 


-312819 


23 


•216126 


•3:^706 


i 24 


•195106 


•29 i271 


24 


•205586 


-312993 


24 


•216307 


•3:35070 


25 


•195279 


•290855 


25 


•205762 


•313:386 


25 


•2164S7 


•3:35482 


26 


•195452 


•29U:39 


23 


•2:5939 


•3137:38 


26 


•216668 


•335795 


27 


-195625 


•291423 


27 


•206115 


•314111 


27 


•216349 


•336157 


28 


•195797 


•2918.16 


23 ! 


•206293 


•314484 


28 


•217030 


•336520 


29 


•195970 


•292190 


29 1 


•206470 


•314856 


29 


•217211 


•336882 


30 


•193143 


•292573 


30 


•206W7 


•315223 


80 


•2173>e 


•337244 


81 


•1913316 


•292956 


31 


•206524 


•315601 


31 


•217578 


•337605 


32 


•196490 


•293339 


82 


•207001 


•315992 


32 


•217754 


•337966 


33 


•190663 


•293721 


33 


•207178 


-316;^44 


38 


•217985 


•338828 


34 


•198836 


•294104 


34 


•207356 


•316716 


84 


•218117 


•338689 


35 


•197009 


•294436 


35 


•207533 


-317087 


35 


•213-299 


•339050 


36 


•197182 


•294363 


36 


•207710 


-317453 


83 


•218480 


•3:39411 


37 


•197356 


•295250 


37 


•207339 


-317529 


87 


•218661 


•339771 


38 


•197530 


•295832 


88 


•203066 


•315200 


38 


•218343 


•340182 


39 


•197703 


•293013 


39 


•208243 


-318571 


39 


•219024 


•340492 


40 


•197877 


•293395 


40 


•208421 


-315942 


40 


•219206 


•840852 


41 


•198051 


•293776 


41 


•203599 


•319:311 


41 


•219:388 


•841212 


42 


•19S225 


•297157 


42 


•208777 


•319682 


42 


•219570 


•3415T8 


43 


•198398 


•297537 


43 


•208954 


-320051 


' 43 


•219752 


•341932 


' 44 


•19S572 


•297918 


44 


•2C9132 


•320421 


44 


•219934 


•342292 


i 45 


•198746 


•29329S 


45 


•209309 


•320791 


i 45 


•220116 


•842651 


46 


•19S930 


•293879 


46 


•209489 


-321161 


46 


•220293 


•343010 


47 


•199 94 


•299059 


47 


•2C9867 


-3-21530 


\ 47 


•220480 


•343369 


48 


•199269 


•2994:39 


48 


•209345 


-321599 


,. 48 


•220662 


•343725 


49 


-192-443 


•299319 


49 


•210023 


•322267 


! 49 


•220844 


•344086 


50 


•199617 


•300193 


50 


•210202 


•322636 


; 50 


•221027 


•844445 


51 


•199792 


•300577 


51 


•210380 


-323005 


i 61 


•221209 


•844808 


52 


•199936 


•30G956 


52 


•210559 


-323373 


52 


•221311 


1 •345161 


53 


•200141 


•301335 


; 53 


•210738 


•323742 


53 


•221574 


1 •845519 


54 


•200315 


•301714 


54 


•21G916 


•324110 


54 


•221757 


•845877 


55 


•200590 


•302093 


, 55 


•211095 


-3-24477 


; 55 


-221940 


•846235 


56 


•2)0665 


•302471 


56 


•211273 


-324344 


56 


•2221-22 


•846592 


57 


•230840 


•302850 


57 


•211452 


•325212 


; 57 


•2-22:305 


•846950 


58 


•201015 


' •303228 


58 


•2116:31 


, •325579 


: 58 


•222488 


•847807 


59 


•201190 


i -303605 


59 


•211310 


•325947 


i 59 


•222671 


•347664 


60 


•201865 


' -303983 


60 


•211990 


•326:314 


■ 60 


•222854 


•343021 



External Secants. 



441 



SSBEGEEES. 


87 DEGREES. | 


S 


8 DEGREES. 




Min. 


Nnt. No. 1 


1 

Lo-arilhm. 


Min. 1 


N;it. No. 


Lo-ariibni | 


Mi;i. 


Nat. No. j 


I.o.-Hrillim 





0-236068 


9-373037 1 





0-252136 


9^401684 





0^269019 


9-429782 


1 


•236380 


•373518 


1 


•252410 


•402107 


1 


•269807 


•430248 




2 


•286591 


•373998 


2 


•252685 


•402579 


2 


•269595 


•430712 ' tl 


8 


•286853 


•374478 


g 


•252960 


•403052 


3 


•269884 


•431178 




4 


•23T115 


•374958 


4 


•253235 


•403525 


4 


•270174 


•431643 




5 


•237377 


•375438 1 


5 


•253511 


•403997 


5 


•270463 


•432108 




6 


•237640 


•3To918 1 
•376897 

•376877 


6 


•258787 


•404469 


6 


•270753 


•482573 




T 


•287932 


7 


•254063 


404941 


1 


•271042 


•433037 




8 


•288165 


8 


•254389 


•4G5418 


8 


•271382 


•433502 




9 


•288423 


•377357 ' 


9 


•254615 


•405884 


9 


•271623 


•433967 




10 


•23SG91 


•377836 j 


10 


•254892 


•406355 


10 


•271914 


•434432 




11 


•23S954 


•37S315 1 


11 


•255169 


•406827 


11 


•272205 


•434896 




12 


•289218 


•378794 ! 


12 


•255446 


•407299 


12 


•272496 


•435301 




13 


•289482 


•379273 1 


13 


•255728 


•407770 


13 


•272788 


•435825 




14 


•289747 


•379752 1 


14 


•2^6000 


•4 8241 


14 


•273080 


•436289 




15 


•240011 


•380231 


15 


•256278 


•4C8711 


15 


•273372 


•436753 




16 


•240276 


•380709 


16 


•256556 


•4G9182 


16 


•273664 


437217 




17 


•240540 


•3S1187 


17 


•256884 


•409653 


" 17 


•273956 


•437680 




18 


•24(1805 


•381666 


18 


•257113 


•4101-24 


18 


•274249 


•438144 




19 


•241070 


•382143 


19 


•257892 


•410595 


19 


•274542 


•438608 




20 


•241335 


•382621 i 


20 


•257671 


•411065 


20 


•274835 


•439072 




21 


•241601 


•383099 


21 


•257950 


•411534 


21 


•2751-38 


•439534 




22 


•241867 


•383577 


22 


•258280 


•412006 


22 


•275421 


•439998 




23 


•242133 


•384055 


23 


•2585C9 


•412475 


28 


•275715 


•440460 




24 


•242400 


•384532 


24 


•258789 


•412946 


24 


•276010 


•440924 




25 


•242567 


•385010 


25 


•259069 


•418415 


25 


•276304 


•441886 




26 


•242933 


•385487 


26 


•259849 


•413884 


26 


•276598 


•441849 




2T 


•243200 


•385964 


27 


•259630 


•414854 


27 


•276898 


•442311 




28 


•243467 


•386440 


28 


•259910 


•414824 


28 


•277188 


•442774 




29 


•248735 


•386918 


29 


•260191 


•415292 


29 


•277584 


•443237 




30 


•244002 


•387394 


30 


•260472 


•415761 


30 


•277780 


•443700 




81 


•244270 


•387871 


31 


•260754 


416231 


31 


•278075 


•444161 




82 


•244539 


•388347 1 


32 


•261035 


•416699 


82 


•278370 


•444623 




33 


•244807 


•888823 ! 


33 


•261317 


•417168 


33 


•278S67 


•445085 




34 


•245075 


•389299 : 


34 


•261600 


•417638 


84 


•278963 


•445547 




35 


•245344 


•389775 


35 


•261882 


•418106 


85 


•279260 


•446009 




86 


•245013 


•390251 


36 


•262165 


•418574 


86 


•279557 


•446471 




8T 


•245882 


•390727 


37 


•262448 


•419043 


87 


•279855 


•446932 




38 


•246152 


•391203 


38 


•262731 


•419511 


88 


•28 '3152 


•447393 




39 


•246422 


•391678 


39 


•268015 


•419980 


89 


•280450 


•447854 




40 


•246691 


•392154 


40 


•263298 


•420447 


40 


•280748 


•448316 




41 


•246961 


•392629 


41 


•268581 


•420914 


41 


•281046 


•448777 




42 


•247273 


•398104 


42 


•268865 


•421382 


42 


•2S1345 


•449239 




43 


•247502 


•393579 


43 


•264150 


•421849 


43 


•281643 


•449699 




44 


•247773 


•394054 


44 


•264435 


422818 


44 


•2S1942 


•450160 




45 


•248044 


•394528 


45 


•264720 


•422785 


45 


•282242 


•450021 




46 


•248315 


•395003 


46 


•265005 


•423253 


46 


•282541 


•451081 




47 


•248587 


•395478 


47 


•265290 


•423720 


47 


•282841 


•451542 




48 


•248859 


•395952 


48 


•265575 


•424187 


48 


•288140 


•-452002 




49 


•249131 


•396427 


49 


•265860 


424653 


49 


•288440 


•452462 




50 


•249403 


•396901 


50 


•266140 


•425120 


5f) 


•288741 


•452922 




51 


•249675 


•397874 


51 


•266432 


425587 


51 


•284042 


•458382 




52 


•249948 


•397848 


52 


•266719 


•426053 


52 


•2S-m3 


•453842 




53 


•250220 


•898821 


53 


•267006 


•426520 


53 


•2S4()44 


•454^02 




54 


•250493 


•398795 


54 


•267298 


•426987 


54 


•284946 


•454762 




55 


•250766 


•399269 


55 


•267580 


•427452 


55 


•285248 


•455222 




56 


•251040 


•399742 


56 


•267867 


•427918 


56 


•285550 


•45r)68l 




57 


•251314 


•400216 


57 


•2681 5 < 


•428384 


57 


•285852 


•456141 




58 


•251588 


•400689 


58 


•268'42 


•428850 


58 


•286154 


•4566(K) 




69 


•251862 


•401101 


59 


•268180 


•429316 


59 


•286457 


•457059 




60 


•252180 


•401634 


60 


•269 J 19 


•429782 


CJ 


•280760 


•457518 





442 








Versed 


Sines. 








39 DEGREES. 


i 40 DEGEEES. 


41 DEGREES. 


Min 




Logaiiihni. 


Mill. 


Nat No 


i Logaii Lni ] 


... 


Nr.t. No 


I..fcaii:hm. 





0-222854 


9-34Sr;21 





0^288955 


^ 98C9188 





0^245291 


9389681 


1 


•223037 


•34S377 


1 


•284143 


•369480 


1 


•245431 


•89001S 


2 


•223220 


•348734 


2 


•234830 


•SC9S27 


2 


•245672 


•89C856 


8 


•223404 


•349C90 


! 3 


•234517 


•870174 


3 


•245864 


•890694 


4 


•223587 


•349446 


4 


•234704 


•87C520 


4 


•246055 


•891031 


5 


•223770 


•349302 


5 


•234S91 


•370867 


5 


•246246 


•891868 


6 


•223954 


•350158 


6 


•285079 


•371213 
•871559 1 


6 


•24W37 


•891705 


T 


•224137 


•350514 


7 


•285266 


7 


•246659 


•892042 ' 


1 8 


•224320 


•35^869 


S 


•235464 


-3719^5 1 


8 


•246S20 


•892379 


i « 


•224504 


•3512-24 


9 


•235641 


•872-251 


9 


•247010 


•892719 


10 


•2246SS 


•851579 


10 


•235S29 


•372597 i 


10 


•2472^2 


•893062 


11 


•224871 


•8519:34 


11 


•286016 


•872942 : 


11 


•247?94 


•89.3388 


12 


•225055 


•8522S9 


12 


•286204 


•873287 : 


12 


•2475S6 


•893T25 


13 


•225239 


•352644 


13 


•286892 


-878682 ! 


13 


•247777 


•894060 


14 


•2254-23 


•852999 


14 


•236580 


•37897S 1 


14 


•24796S 


•894896 


15 


•225607 


•853853 


15 


•236768 


•874322 1 


15 


•248160 


•894732 


16 


•225791 


•858707 ; 


16 


•236956 


•874667 1 


16 


•248352 


•895C67 


IT 


•225976 


•854062 ; 


17 


•287144 


•875011 ; 


17 


•248544 


•895403 


18 


•226160 


•854415 '• 


18 


•287332 


•875355 1 


18 


•248786 


•895T88 


19 


•226344 


•854769 : 


19 


•237520 


•375700 1 


19 


•248928 


•896074 


20 


•22652S 


•355122 


20 


•287708 


•376044 ! 


20 


•249120 


•896408 


21 


•226723 


•855476 


21 


•237897 


-876388 1 


21 


•249312 


•896743 


22 


•226898 


•355830 


22 


•238^85 


•376731 1 


22 


•249504 


•897078 


23 


•227082 


-856183 


23 


•288273 


•877075 i 


23 


•249696 


-897412 


24 


•227266 


•356535 


24 


•288462 


•877413 


24 


•249889 


•897T47 


25 


•227451 


•856888 


25 


•238650 


•377762 


25 


•250082 


•398082 


26 


•227636 


•357241 


26 


•238839 


•878105 i 


26 


•250275 


•398416 


2T 


•227821 


•357593 


27 


•289028 


-878448 i 


27 


•250466 


•898749 


2S 


•228005 


•857945 


28 


•289-216 


•378790 i 


28 


•250659 


•899083 


29 


•22S190 


•858297 


29 


•2S9405 


■879183 1 


29 


•250852 


•889417 


80 


•228375 


•358650 


30 


•289594 


•379476 i 


30 


•251045 


•899751 


31 


•228560 


•359001 ' 


31 


•289783 


•379818 i 


31 


•251237 


•400083 


32 


•228746 


•859353 


82 


•239972 


■8S0160 1 


32 


•251430 


•400417 


33 


•22S931 


•359704 


33 


•240162 


•8S0503 • 


33 


•251623 


•400750 


84 


•229116 


•860056 


84 


•240851 


•83f845 ! 


34 


•251816 


•401(83 


So 


•229301 


•86C407 


85 


•240540 


•381186 


35 


•2520C9 


•401415 


36 


•229486 


•360757 


36 


•240729 


•8S152S : 


86 


•2522C2 


•401748 


37 


•229672 


•861108 


37 


•240919 


•S81S70 


37 


•252395 


•402081 


38 


•229857 


•861459 


38 


•«41108 


•832211 1 


38 


•252588 


•402413 


39 


•230048 


•861 SC9 


89 


•241297 


•3S2552 


89 


•252782 


•402745 


40 


•230229 


•362160 


40 


•241486 


•382892 ■ 


40 


•252975 


•403077 


41 


•230415 


•362510 


41 


•241676 


•388233 ; 


41 


•253168 


•4034(9 


42 


•230600 


•362S60 


42 


•241866 


•383574 1 


42 


•253362 


•403741 


43 


•230786 


•363210 


43 


•242056 


•833915 


43 


•253566 


•4C4CT3 


44 


•23C.972 


•363559 


44 


•242-245 


•834255 


44 


•253749 


•404404 


45 


•231158 


•363909 


45 


•242485 


•884595 ! 


45 


•253942 


•404785 


46 


•231344 


•864259 


46 


•242625 


•8S4935 • 


46 


•254136 


•405067 


47 


•231580 


•86460S 


47 


•242815 


•885275 : 


47 


•254330 


•405398 


48 


•231716 


•364957 


48 


•243005 


•885615 : 


48 


•254524 


•405729 


49 


•231903 


•865306 


49 


"248195 


-335955 j 


49 


•254718 


•406060 


50 


•232G89 


•365654 


50 


^ -248885 


•386294 


60 


•254912 


•406390 


51 


•232275 


^866003 


61 


•243575 


•386633 


61 


•255106 


•406721 


62 


•232461 


•866351 


62 


•248766 


•336973 


52 


•255300 


•407051 


53 


•232649 


•366699 


53 


•243956 


•387812 


63 


•255494 


•407881 


64 


•232S35 


•367048 


54 


•244146 


•887650 ' 


54 


•255689 


•407711 


55 


•233021 


•867896 


65 


•244387 


•3879S9 ! 


65 


•2558S-3 


•4(8041 


66 


•233209 


•367744 


66 


•244528 


•388328 


66 


•256077 


•408371 


67 


■233895 


•369991 


67 


•244719 


•883667 


67 


•256272 


•4C8701 


6S 


•233582 


•368489 


68 


•244909 


■889004 i 


58 


•256466 


•409G3O 


69 


1 ^233769 


•368786 


69 


! -245100 


•889343 1 


69 


•256660 


•400359 


m 


i -233955 


•869133 




•245-291 

1 


•389681 


60 


•256855 


•409688 



External Secants. 



443 



89 DEGEEES. 


40 DEGREES. 


41 DEGEEES. 


Min. 


j Nat. No. 


1 Logarithm 


Min. 


1 Nat. No. 


i Logarithm. 


Min. 


1 Nat. No. 


1 L..„'u-it!mi. 





0-2S6760 


9-457518 





0^305407 


9^484879 


i « 


0-325013 


9-511931 


1 


•2S7063 


•457977 


1 


•305726 


•485332 


1 


•325343 


-512348 


2 


•287366 


•458436 


2 


•306045 


•4S5785 


2 


♦3256S4 


•512796 


8 


•287670 


•458895 


3 


•306364 


•486238 


3 


•326020 


•513244 


4 


•237974 


•459353 


4 


•306684 


•486691 


4 


•326355 


•513C91 


5 


•288279 


•459312 


5 


•307004 


•487144 


5 


•326692 


•514138 


G 


•238583 


•460270 


6 


•307324 


•487597 


G 


•327029 


-514535 


7 


' -283883 


' -460729 


r 


•307644 


•488049 


7 


•327365 


•515032 


8 


•289193 


•461187 


8 


•307935 


•488501 


8 


•327702 


•515483 


9 


•2S949S 


•461645 


9 


•308283 


•483954 


9 


•328040 


•515926 


la 


•2898 J3 


•462102 


10 


•308607 


•489406 


10 


•3-28378 


•516374 


11 


•29J109 


•462560 


11 


•308928 


•489358 


11 


•328716 


•516820 


12 


•29:)415 


•463018 


12 


•309250 


•493310 


12 


'329355 


•51726S 


13 


•293721 


•463476 


13 


•309572 


•493761 


13 


•329393 


•517714 


14 


•291023 


•463934 


14 


•309394 


•491214 


14 


•329731 


•513160 


15 


•291385 


•464392 


15 


•310217 


•491665 


16 


•330070 


•518607 


1C> 


•291G41 


•464349 


16 


•310540 


•492117 


16 


•83)410 


•519053 


ir 


•291949 


•465307 


17 


•310863 


•492568 


^17 


•33)750 


•519499 


IS 


'292257 


•465764 


18 


•311186 


•493019 


18 


•331093 


'519945 


VJ 


•2325G4 


•466221 


19 


•311510 


•493471 


19 


•331430 


•520392 


20 


•292372 


•46GG78 


20 


•311834 


•493923 


20 


•331770 


•520837 


21 


•293181 


•467135 


21 


•312158 


•494374 


21 


•332111 


•521284 


22 


•29349) 


-467593 


22 


.312482 


•494824 


22 


•332452 


•521730 


23 


•203T98 


•468)49 


23 


•3128J7 


•495276 


23 


•332794 


•522175 


24 


•291107 


•468505 


24 


'313132 


•495726 


24 


•333136 


•522621 


25 


•2;)U1T 


•463932 


25 


•313457 


•496178 


25 


•333478 


•523367 


'2j 


•JJi:ii7 


•4G0419 


26 


•313782 


•496628 


26 


•333320 


•52)513 


2T 


•293 ;36 


•469375 


27 


'314108 


•497079 


27 


•334163 


•523953 


23 


•293346 


•470331 


23 


•314434 


•497529 


28 


•334506 


•524403 


29 


•295656 


•470787 


29 


•314760 


•497930 


29 


•334849 


•524849 


30 


•295937 


■471244 


30 


•315086 


•498430 


80 


•335193 


•525295 


81 


'2962T8 


•471699 


81 


•315413 


'493830 


81 


•335536 


•525739 


32 


•29J589 


•472155 


82 


•315740 


•499330 


32 


•335380 


•523185 


83 


•2969J0 


•472611 


33 


•316068 


•499781 


83 


•336324 


•526629 


84 


•297212 


•473J67 


34 


•316393 


•500231 


34 


•336569 


•527074 


85 


•29T524 


•473522 


35 


•316724 


•500681 


85 


•336914 


•527519 


36 


•297333 


•473977 


36 


•317052 


•501131 I 


86 


'337260 


•527934 


3T 


•293143 


•474132 


87 


•317331 


•501581 1 


87 


'337605 


•5284C9 


88 


•293461 


•474333 


38 


'317710 


'502031 ; 


88 


•337930 


•523353 


89 


•298774 


-475343 


89 


•318339 


•502430 1 


89 


•33S29G 


•529297 


4t) 


•299, )S3 


•475793 


40 


•318)38 


•502929 j 


40 


•333643 


•529742 


41 


•2994)1 


-476253 


41 


•318697 


'503378 1 


41 


•338990 


•530186 


42 


•299715 


•476708 


42 


•319,27 


•503328 ' 


42 


•339337 


•530631 


43 


•300029 


•477163 


43 


•319356 


•504277 1 


43 


•339034 


•531075 


U 


•300343 


•477617 


44 


•319387 


•504726 ' 


44 


•340031 


•631519 


45 


•800658 


•478)72 


45 


•320018 


•505175 , 


45 


'340379 


•531963 


46 


•300973 


•478527 


46 


•320350 


•505624 


46 


•340723 


•532408 


47 


•301283 


•478981 


47 


•320681 


•506073 


47 


•341077 


•532351 


48 


•301603 


•479435 


48 


•821013 


•506522 


48 


•341425 


•533295 


49 


•301918 


•4793S9 


49 


•321345 


•506971 


49 


•341774 


•533739 


50 


•302234 


•480341 


50- 


•821677 


•507419 


60 


•342123 


•5:54183 


51 


•302550 


•480798 


61 


•822309 


•507867 


51 


•842473 


•534626 


52 


•302866 


•481231 


62 


•322342 


•50S317 


62 


•342823 


•535370 


53 


•303183 


•481705 


63 


'322375 


•508765 1 


63 


•343173 


•533513 


54 


•303500 


•482159 


64 


•323003 


•509212 


54 


•343523 


•5^3956 


55 ; 


•303818 


•482613 


55 


'823341 


•509661 


65 


•343874 


•53(5400 


56 


•304135 


•483066 


5() 


•823675 


•510109 


66 


•344226 


•536843 


57 


•304152 


•483519 


57 


•3240(19 


-510558 


57 


•844578 


•537287 


58 


•304770 


•483973 


58 


•324343 


•511005 


58 


•344929 j 


•537729 


69 


•805f)s9 


•484426 


59 


•324678 


•611453 


69 


•843231 


•538172 


60 


•305407 


•484879 


60 


•325013 


•611901 


CO 


•845633 ! 


•533615 • 









444 



Verskd Sines. 





12 DEGEEES. 


|| 


43 DEGEEES. 




44 DEGEEES. 




1 Nat. No. 


j Logariihm. 


l! Min 


1 Nat. Nc. 


j Logarithm. 








Mill. 


Min. 


1 Nat. No. 


Logarithm. 





1 0-256855 


9-4C9381 


i 


■ C •238046 


i 9^429181 





' 0-280660 


9-448181 


1 


i -257050 


•410017 


1 1 


' •268815 


1 ^429502 


1 


: -280862 


•448493 


2 


! '257245 


-41G346 


i 2 


, -2(^.43 


1 ^429822 


2 


•281064 


•448806 


3 


i -257489 


, -410674 


! 3 


•269242 


•430142 


3 


•281266 


•449118 


4 


i -257634 


i -411003 


4 


i -269440 


•430463 


4 


•281469 


•449431 


5 


-257829 


-411332 


5 


! •269689 


i -430783 


5 


•281671 


•449743 


6 


-258024 


•411660 


6 


■2C9S38 


•4811C3 


6 


•281874 


•450055 


7 


-25S219 


•4119S8 


7 


•270037 


•431423 


7 


•2S2076 


•450366 


8 


! -258414 


•412317 


1 8 


•27C236 


•431743 


8 


•282279 


•450678 


9 


•25S6C9 


•412644 


1 9 


-270434 


•432061 


9 


•282482 


•450990 


10 


-258805 


•412972 


; 10 


•27C6S3 


•432381 


10 


•282684 


•451301 


11 


-259000 


•413299 


1 11 


-270833 


•432701 


11 


•282387 


•451613 


12 


■259196 


•413628 


! 12 


•271032 


•433020 


12 


•283090 


•451924 


13 


-259391 


•41S955 


I 13 


-271231 


■438339 ' 


13 


•283298 


•452235 


14 


-2595S6 


•4142S2 


14 


-271430 


•433657 


14 


•283495 


•452545 


15 


-259782 


•414609 


15 


•271629 


•433976 ! 


15 


•283698 


•452856 


16 


•259977 


•414935 


16 


•271828 


•434295 ; 


16 


•233901 


-453167 


17 


-260173 


•415262 


17 


•272028 


•434613 


17 


•284104 


•458477 


18 


•260369 


•415SS9 


18 


•272227 


•434931 


18 


•284307 


•453788 


19 


-260565 


•415916 ; 


19 


•272427 


•435250 i 


19 


•284510 


•454098 


20 


•260760 


•416242 1 


23 


•272626 


•435568 1 


20 


•284713 


•454408 


21 


-260956 


•416568 1 


21 


•272826 


•435S86 ; 


21 


•284917 


•454719 


22 


•261152 


•416894 ; 


22 


•273C28 


•436204 : 


22 


•285120 


•455C28 


23 


•2613^18 


•417220 ; 


23 


•27S225 


•436521 ; 


23 


•285324 


•455338 


24 


•261544 


•417546 1 


24 


•2T3425 


•436889 ; 


24 


•285527 


•455647 


25 


•261740 


•417871 ! 


25 


•273625 


•437156 : 


25 


•285731 


•455957 


26 


•261937 


•418197 I 


26 


•273825 


•437473 1 


26 


•285934 


•456266 


27 


•262133 


•418522 1 
•418847 1 


27 


•274025 


•437790 1 


27 


•286138 


•456576 


28 


•262380 


28 


•274225 


•438107 i 


28 


•286842 


•456835 


29 


-232526 


•419172 ! 


29 


•274425 


•438424 ' 


29 


•286546 


•457194 


30 


•262722 


•419497 i 


30 


•274625 


•438741 I 


30 


•236750 


•457503 


31 


•262919 


•419322 i 


31 


•274825 


•439057 ; 


31 


•2369.58 


•457811 


32 


•283116 


•420147 1 


32 


-275026 


-439374 i 


32 


•287157 


•458120 


33 


•263313 


•420472 i 


33 


•275226 


•439690 : 


33 


•287861 


•453428 


34 


•263510 


•420795 ! 


34 


•275427 


•440006 


34 


•287505 


•458736 


35 


•263706 


•421120 1 


35 


•275623 


-440323 : 


35 


•287770 


•459045 


36 


•263903 


•421444 i 


36 


•275828 


••440639 : 


86 


•287974 


•459853 


37 


•264100 


•421768 ' 


37 


•276029 


•44t:954 i 


37 


•2831 78 


•459661 


38 


•264297 


'422-m 1 


38 


•276230 


•441270 i 


38 


•288883 


•459969 


39 


•264494 


•422416 


39 


•276430 


•441585 . 


89 


•238587 


•460277 


40 


•261691 


•422739 


40 


•276631 


•441901 ; 


40 


•283792 


460585 


41 


•264888 


•423062 


41 


•276832 


•442216 j 


41 


•288998 


•460892 


42 


•265086 


•423386 ; 


42 


•277038 


•442531 1 


42 


•289200 


•461199 


43 


•265233 


•423709 : 


43 


•277234 


•442846 : 


43 


•289405 


•461506 


44 


•265480 


•424032 


44 


•277435 


•443161 i 


44 


•289610 


•461813 


45 


•265677 


•424354 


45 


•277636 


•443475 ; 


45 


•239314 


•462120 


46 


•265875 


•424677 


46 


•277837 


•443790 : 


46 


•290019 


•462427 


47 


•266072 


•424999 i 


47 


•278039 


•444105 \ 


47 


•29G224 


•462734 


48 


•26C270 


•425322 


48 


-278240 


•444419 , 


48 


•290429 


•463041 


49 


•266468 


•425644 


49 


•278441 


•444734 


49 


•290634 


•463347 


50 


•266666 


•425967 


50 


•278642 


•44504X 


50 


•29C839 


•463653 


51 


•266833 


•426289 i 


51 


•278844 


•445361 


51 


•291044 


•463959 


52 


•267061 


•426611 1 


52 


•279045 


•445675 i 


52 


•291249 


•464265 


53 


•267259 


•426932 ' 


53 


•379247 


•4459S9 : 


53 


•291454 


•464571 


54 


•267457 


•427254 ! 


54 


•279448 


•446802 ! 


54 


•291660 


•464377 


55 


•267655 


•427576 ! 


55 


•279650 


•446615 i 


55 


•291365 


•465182 


56 


•267853 


•427897 


56 


•279852 


•446929 i 


56 


•292070 


•465488 


57 


♦268052 


•428219 i 


57 


•280054 


•447242 j 


57 


•292276 


•465794 


58 


•268250 


•428539 


58 


•2S0256 


•447555 


58 


•292482 


•466099 


59 


•268448 


•42S860 


59 


•280453 


•447868 


59 


•292688 


•466404 


60 


•268646 


•429181 


60 


•280660 


-448181 


60 


•292893 


•466709 



External Secants. 



445 





42 DEGREES. 


1 

1 


43 DEGEEES. 


I 


44 DEGREES. 


Min. 


Nat. No. 


Lo^aritlim. 


! Min. 


Nat. No. 


Logarithm. 


M.n. 


Nat. No. 1 Logarithm. 





0-345G33 


9-588615 


i 0' 


0^367328 


9^565C54 





0-890164 9 


591247 


1 


•345985 


•589;-57 


1 1 


•867C99 


•565492 


1 


•89G554 


591681 


2 


•346388 


•589500 


2 


•368070 


•565930 


2 


•89C945 


692116 


3 


•346G91 


•539942 


3 


•368441 


•566368 


3 


•891886 


592550 


4 


•34T044 


•540385 


4 


•868813 


•566807 


4 


•891728 


692985 


6 


•847809 


•540828 


5 


•8691S6 


•567245 


5 


•892121 


593420 


6 


•347753 


•541270 


6 


•369559 


•567684 


6 


•892513 


598854 


7 


•348107 


•541712 


7 


•369982 


•568122 


7 


•392905 


594288 


8 


•348462 


•542156 


8 


•370805 


•568560 


8 


•898298 


694722 


9 


•848817 


•542597 


9 


•870678 


•568997 


9 


•898692 


695157 


10 


•849172 


•543039 


I 10 


•871052 


•509435 


10 


•894085 


595590 


11 


•349528 


•543482 


i 11 


•371427 


•569674 


11 


•894479 


596024 


12 


•349884 


•548924 


: 12 


•371801 


•570311 


12 


•894874 


596459 


13 


•350240 


•544866 


13 


•372176 


•570749 


13 


•896269 


696S93 


14 


•350597 


•544808 


14 


•372651 


•571186 


14 


•895664 


597326 


15 


•350954 


■545249 


15 


•872926 


•571623 


15 


•896069 


597760 


16 


•351310 


•545690 


16 


•373302 


•572061 


16 


•396555 


598194 


17 


•351667 


•546132 


17 


•378679 


•572498 


-17 


•896861 


698627 


18 


•852025 


•546574 


18 


•374055 


•672935 


18 


•897247 


699061 


19 


'352384 


•547016 


19 


•374432 


•573373 


19 


•897644 


699495 


20 


•852742 


•547457 


1 20 


•374809 


•578810 


20 


•898041 


699928 


21 


'353100 


•547898 


1 21 


•375187 


•574248 


21 


•398489 


60C862 


22 


'353459 


•548839 


1 22 


•375565 


•574685 


22 


•898837 


600795 


23 


•353818 


•548780 


1 23 


•375943 


•575122 


23 


•899285 


601229 


24 


•354178 


•549222 


24 


•876321 


•575559 


24 


•899683 


601661 


25 


•354538 


•549663 


25 


•376700 


•575995 


25 


•400082 


6C2C95 


26 


'354S98 


•550104 


26 


•377079 


•576432 


26 


•40C431 


602528 


27 


•355258 


•550544 


27 


•377478 


•576868 


27 


•400881 


6C2962 


28 


•355619 


•550985 


28 


•877838 


•577806 


28 


•4C1231 


608895 


29 


•355980 


•551425 


29 


•378218 


•577742 


29 


•401681 


608828 


30 


•356341 


♦551866 


30 


•878599 


•578179 


30 


•4C2G32 


604261 


31 


'356703 


•552307 


31 


'378979 


•578615 


31 


•402483 


604693 


32 


'357065 


•552748 


32 


•379360 


•579C52 


82 


•402834 


605126 


33 


•357427 


•553189 


33 


•879741 


•579488 


83 


•4C8285 


606559 


34 


•357790 


•653629 


34 


•380123 


•579924 


84 


•408637 


6(6991 


35 


•358153 


•554069 


35 


•380505 


•580861 


85 


•404040 


6064'25 


36 


•358516 


•554509 


36 


•380888 


•580797 


36 


•404443 


606857 


37 


•358880 


•554949 


37 


•381270 


•581233 


87 


•4C4846 


607290 


38 


•359244 


•555889 


38 


•881658 


•581669 


88 


•4C5249 


607722 


39 


•359608 


•555829 


39 


•382037 


•582105 


39 


•4(]5668 


6(8155 


40 


•859972 


•556269 


40 


•382420 


•582541 


40 


•406068 


608588 


41 


•800337 


•556709 


41 


•382804 


•582977 


41 


•406462 


6(9020 


42 


•360702 


•557149 


42 


•383188 


•583412 


42 


•406867 


6(9452 


43 


•361068 


•557589 


43 


•388573 


•588848 


43 


•407272 


6(9884 


44 


•361433 


•558028 


44 


•388958 


•584284 


44 


•407678 


610816 


45 


•361799 


•558467 


45 


•384343 


•584719 


45 


•408084 


610748 


46 


•362166 


•558907 


46 


•384729' 


•585155 


46 


•408490 


611181 


47 


•362522 


•559846 


47 


•385115 


•685591 


47 


•408996 


611613 


48 


•862899 


•559786 


48 


•385501 


•586026 


48 


•4C9403 


612045 


49 


•368266 


•560225 


49 


•385888 


•586462 


49 


•409710 


612477 


50 


•803684 


•560665 


50 


•386275 


•586896 


50 


•410117 


612908 


51 


•364002 


•561104 


51 


•386662 


•587332 


51 


•410525 


618340 


52 


•364870 


•561543 


52 


•387050 


•587767 


52 


•410934 


613772 


53 


•364789 


•561982 


53 


•887488 


•588203 


53 


•4113-12 


614208 


54 


•365108 


V ^562421 


54 


•887826 


•588637 


54 


•411751 


614685 


55 


•365477 


•562860 


55 


•888215 


•589702 


55 


•412160 


615066 


56 


•365847 


•563299 


56 


•388604 


•589507 


56 


•412570 


615498 


57 


•366217 


•563788 


57 


•888993 


•589942 


57 


•412981 


615930 


58 


•366587 


•564176 


58 


•389883 


•590877 


58 


•418391 


616861 


59 


•866957 


•564616 


59 


•889773 


•590812 


59 


•418802 


616793 


60 


•367328 


•565054 


60 


•390164 


•591247 


60 


•414218 


617224 



88 



416 






Vj-rsed S 


IX E3. 














45 DEGEEES. 


43 DEGEEES. 


47 DEGEEES. 


... 1 


Nat No. 


Loga ithm 


Min. 


Nat No. 


LogHiit. m 1 


Min. 1 


Nat No 1 


Logarilhm. | 





0-292S98 


9^466709 j 
467014 i 





©•305342 


9-484786 


i 


0-318001 


9-502429 1 


1 


•293099 


1 


•305551 


•485083 


1 


•318214 


•50272} 


2 


•293305 


467319 


2 


•305760 


•485381 


2 


•318427 


•503010 i 


3 


•293511 


467624 


3 


•305970 


•485678 


3 


•318640 


•503300 


4 


•293717 


467928 


4 


•306180 


•435976 


4 


•318853 


•503590 


5 


•293923 


•468233 


5 


•306339 


•486273 


5 


•319066 


•503880 


6 


•291129 


•463537 


6 


•306593 


•486570 


6 


•319279 


•5C4170 


7 ; 


•294335 


•463342 


7 


•306308 


•486366 


7 


•319492 


•504460 


8 1 


•29i541 


•469146 


8 


•307017 


•4S7163 


8 


•819705 


•504750 


9 I 


•294747 


•469 U9 


9 


•307227 


•437460 


9 


•319918 


•5051)39 


10 j 


•294953 


469753 


10 


•307437 


•487757 


j 10 


•320182 


•505329 


11 ! 


•295159 


•470056 


It 


•307647 


•488053 


i 1^ 


•320845 


•505618 


12 


•295385 


•470860 


12 


•307857 


•488349 


12 


•820558 


•505907 


13 


•295572 


•470633 


13 


•308067 


•488645 


13 


•820772 


•506196 


14 


•295778 


•470936 


14 


•308277 


•483941 


14 


•320986 


•506486 


15 


•2959S5 


•471270 


15 


•308437 


•4S9237 


15 


•821200 


•506775 


16 


•293192 


•471573 


16 


•308697 


•489532 


16 


•321414 


•507064 


IT 


•293393 


•471376 


17 


•308907 


439828 


1 17 


•321627 


•507352 


18 


•293605 


472179 ^ 


18 


•309118 


•4901-24 


i 18 


•821841 


•507641 


19 


•293812 


•472431 


19 


•309323 


•490419 


! 19 


•322054 


•507929 


2,-) 


•297019 


•472784 i 


20 


•309538 


•490714 


1 20 


•322268 


•50821T 


21 


•297226 


•473087 


21 


•309748 


•4910C9 


' 21 


•322482 


•508505 


22 


•297433 


•473389 ' 


22 


'309959 


491305 


1 22 


•322696 


•508793 


23 


•297640 


•478691 


23 


•310170 


•491600 


1 23 


•322910 


•509081 


24 


•297847 


•473993 . 


24 


•310381 


•491895 


: 24 


•323124 


•509369 


25 


•298054 


•474295 : 


25 


•310591 


492189 


i 25 


•323333 


•509657 


26 


•29^261 


474597 


26 


•310801 


•492483 


• 26 


•323552 


•509944 


27 


•298468 


•474398 


27 


•311012 


•492778 


; 27 


•323766 


•510231 


23 


•293676 


475200 


28 


•311223 


•493072 


, 28 


•323981 


•510519 


29 


•293833 


•475502 


29 


•311134 


•493366 


: 29 


•324196 


•510807 


80 


•299 J91 


•4753 J3 


30 


•311645 


•493660 


I 30 


•324410 


•511094 


81 


•299298 


•476104 


31 


•311856 


•493955 


31 


•824624 


•511381 


32 


•299506 


•476405 


32 


•312068 


494219 


: 32 


•324889 


•511668 


33 


•299713 


•476706 


33 


•312279 


•494543 


83 


•325053 


•511955 


34 


•299921 


•477U07 


34 


•312490 


•494836 


84 


•3-25268 


•512-242 


85 


•300129 


477308 


35 


•312702 


•495130 


1 35 


•825483 


•512529 


36 


•300337 


•477609 


86 


-312913 


•495423 


i 36 


•825693 


•512815 


37 


•300545 


•4779.9 1 


87 


•313124 


•495716 


' 37 


•825912 


•513101 


38 


•800752 


•4782:9 


38 


•313335 


•496009 


38 


•826127 


•5133S7 


39 


•300960 


•473509 


39 


•313547 


•496302 


39 


•823842 


•513673 


40 


•301168 


•473309 


40 


•313759 


•496596 


40 


•326557 


•513959 


41 


•301376 


479109 


41 


•318970 


•496838 


: 41 


•326772 


•514245 


42 


•301584 


•479409 


42 


•3141S2 


497181 


42 


•826987 


•514531 


43 


•301793 


479709 


43 


•314393 


•497474 


; 43 


•327203 


•614817 


44 


•302;X)1 


•480008 


44 


•314605 


•497766 


' 44 


•827418 


•515102 


45 


•302210 


•480308 


45 


•314817 


493058 


\ 45 


•327633 


•515388 


46 


•302418 


•433607 


46 


•315029 


•493350 


i 46 


•327848 


•515673 


47 


•302626 


430907 


47 


•315-241 


•498642 


i 47 


•328064 


•515958 


48 


•3'J2335 


•481206 


48 


•315453 


•493934 


48 


•328230 


•516244 


49 


•303043 


•481505 


49 


•315665 


•409226 


49 


•823495 


•516529 


5a 


•303252 


•481304 


50 


•815377 


•499518 


1 50 


•328711 


•516314 


51 


•303461 


•482102 


51 


•316089 


•499809 


I -51 


•325926 


•517C98 


52 


•303670 


•482401 


52 


•316301 


•500101 


i 52 


•329142 


•517383 


53 


•303878 


•432699 


1 53 


•316513 


•500392 


: 53 


•329358 


•517668 


54 


•334087 


•482993 


i 54 


•316726 


•500684 


1 54 


•829573 


•517952 


55 


•304296 


•483296 


55 


•316939 


•500976 


j 55 


•329789 


•518236 


56 


•304505 


•433595 


1 56 


•317152 


•501267 


56 


•330005 


•518521 


57 


•304714 


•483893 


57 


•317864 


•501557 


57 


•330221 


, •518S05 


58 


•304923 


•484191 


58 


•317576 


•501848 


58 


•33C437 


•519C89 


59 


•305132 


•484488 


59 


•317789 


•502139 


59 


•330653 


•519378 


60 


•805342 


•484736 


60 


•318901 


•502429 


60 


•83C869 


•519656 



External SecxVNts. 



44*7 



45 DEGREES. 




46 DEGREES. | 


47 DEGREES. 


Min 


N.-i(, No. 


Logarithm. 


Min. 


Nat No 1 L 


igarithm | 


Min. 


Nat. No. 


Lojarithtn | 





0-414213 


9-617224 


i 


0-439557 9 


643; ;15 





0^466279 


9-668646 ; 


1 


•414G25 


•617655 


i 1 


•439991 


643444 


1 


•466737 


•669072 1 


2 


•415:^37 


•618087 


1 2 


•440425 


643872 


2 


•467195 


•669493 ' 


3 


•415450 


•613518 


! 3 


•440859 


644300 


8 


•467653 


•669924 


4 


•415863 


•61S950 


4 


•441294 


644728 


4 


•4GS112 


•670350 


5 


•416276 


•G193S1 


5 


•441729 


645156 


5 


•463571 


•670776 


6 


•416689 


•019311 


6 


•442164 


645584 


6 


•4G9::;30 


•671201 


7 


•417102 


•620242 


7 


'442600 


646012 


7 


•469490 


•671627 


8 


•417516 


•620673 


8 


•44:3037 


646440 


8 


•469951 


•672^53 


9 


•417920 


•621(94 


! 9 


•443475 


646869 


9 


•470412 


•672478 


10 


•413345 


•621535 


i 10 


•44:^912 


647-297 


10 


•47C373 


•672904 


11 


•418760 


•621965 


1 11 


•444350 


647725 


11 


•471335 


•678880 


12 


•419176 


•622396 


! 12 


•444783 


643153 


12 


•471797 


•673755 


13 


•419592 


•622327 


i 13 


•445226 


643581 


13 


•472260 


•674181 


14 


•420008 


•623257 


1 14 


•445665 


649)09 


14 


•472723 


•674607 


15 


•420425 


•623688 


1 15 


•446105 


649437 


15 


•473187 


•675033 


16 


•420842 


•624119 


i 16 


•446544 


649364 


16 


•473650 


•675458 


17 


•421259 


•624^49 


i 17 


•446984 


65U292 


17 


•474114 


•675383 


18 


•421677 


•62493a 


1 18 


•447425 


650720 


18 


•474579 


•6763C9 


19 


•422095 


•625410 


' 19 


•447865 


6.M147 


19 


•475044 


•676734 


20 


•422513 


•625840 


1 20 


•448306 


651574 


20 


•4755r.9 


•677159 


21 


•422932 


•626271 


i 21 


•44S748 


052002 


21 


•475975 


•677584 


22 


•423351 


•626701 


22 


•449190 


'o5243C 


22 


•476442 


•678010 


23 


•423771 


•627131 


23 


•449632 


652357 


23 


•476908 


•678485 


24 


•424191 


•627561 


24 


•450075 


653285 


24 


•477375 


•678860 


25 


•424611 


•627991 


25 


•450518 


653712 


25 


•477343 


•679-285 


26 


•425031 


•623421 


26 


•450961 


654139 


26 


•478311 


•679710 


27 


•425452 


•628351 


27 


•451405 


654567 


27 


•478779 


•689185 


28 


•425874 


•629281 


28 


•451850 


651994 


28 


•479248 


•680560 


29 


•426296 


•629711 


29 


•452294 


655421 


29 


•479718 


•680936 


80 


•426718 


•630141 


80 


•4527S9 


655848 


80 


•480133 


•631411 


31 


•427141 


•630571 


31 


•453135 


656276 


81 


•43:)658 


•631836 


82 


•427563 


•C31000 


32 


•453631 


656703 


82 


•481129 


•683361 


33 


•4279S6 


•631430 


83 


•454 ,77 


657130 


88 


•431600 


•633636 


34 


•42S110 


•631880 


1 84 


•454524 


657557 


84 


•432,;71 


•633111 


35 


•423835 


•632290 


85 


•454971 


6572 t 


35 


•432543 


•683536 : 


36 


•429260 


•63-372) 


86 


•455419 


6534U 


86 


•453015 


•6SS93) 


37 


•429684 


•633149 


87 


•455867 


653838 


87 


•433487 


•634:335 


33 


•43.1 9 


•633578 


88 


•456315 i 


659265 


83 


•483960 


•6843 D 


89 


•43:)534 


•6340C8 


39 


•45G764 


659o92 


89 


•484433 


•635334 1 


40 


•43 970 


•634487 


1 40 
1 41 


•457313 


66J119 


40 


•4S49.)7 


•685653 i 


41 


•431:386 


•634866 


•457662 


660545 


41 


•435381 


•636,83 i 


42 


•431312 


•635295 


1 42 


•458112 


66;.972 


42 


•435356 


•636508 1 


43 


•432239 


•035724 


1 43 


•458562 


661398 


43 


•436332 


•686933 i 


44 


•432667 


•63G154 


44 


•459013 


661825 


44 


•4363'. 8 


•637353 i 


45 


•433,95 


•636583 


45 


•459464 


6G2351 


45 


•437284 


•687783 : 


46 


•433523 


•637;,12 


46 


•459915 


662678 


46 


•487760 


•683306 ' 


47 


•433951 


•637441 


47 


•460367 


663104 


47 


•4S3337 


•63S331 


48 


•434380 


•G37870 


48 


•460820 


663531 


48 


•433714 


•689:55 i 


49 


•434310 


•638399 


49 


•461273 


663958 


49 


•439192 


•689479 i 


50 


•435339 


•638728 


50 


•461726 


664:384 


50 


•4S9G70 


•639904 


51 


•4356G9 


•639157 


51 


•462179 


664310 


51 


•40J149 


•691328 


53 


•436100 


•639586 


52 


•462632 


665236 


52 


•493623 


•69J752 


53 


•4:^6530 


•G40014 


53 


•463087 


665663 


53 


•4911; 8 


•691177 


54 


•430901 


•640443 


54 


•463542 


666;)89 


54 


•49158S 


•691601 


55 


•437::9 i 


•64,872 


55 


•468993 


666516 


55 


•493069 


•692:26 


56 


•437825 


•641301 


56 


•464458 


666942 


56 


•493550 


•693450 


57 


•4:-5S258 


•641 730 


57 


•464906 


667368 


57 


•493J31 


•693874 


58 


•43SC9:) 


•642158 


58 


•465365 


•667794 


5S 


•49::>512 


•69:)393 


50 


•430123 


•6425S6 


59 


•465S22 


•6i:S32) 


59 


•49:3904 


•693722 


60 


•489557 


•Gi3Jl5 


60 


•466279 


•663646 


G) 


•494477 


•694146 



448 








Versed 


Sines. 








48 DEGPwEES. 


[ 


49 DEGEEES. 


50 DEGREES. 


' Min. 


Nat No. 


Logaiithm. 


Min 


Nat No 


/...gariibm. 


Min. 


Nat No 


Lopariibm. 





0-330869 


9-519656 





0-343941 


9-536454 ' 





0-357213 


9-552927 


1 


•331085 


•519940 


1 


-344160 


•536761 1 


1 


-357435 


•553197 


2 


•331302 


•5202-24 


2 


•344:380 


•537038 i 


2 


•357658 


•553468 


3 


•3:31518 


•520507 


3 


•344600 


•537315 ! 


3 


•3578S1 


-553789 


4 


•331735 


•520791 


4 


•344820 


•537592 ; 


4 


•358104 


•5540(9 


5 


•331951 


•521074 


5 


-345039 


•5:37863 , 


5 


•858327 


•554280 


; 6 


•3:3-2167 


•521357 


6 


-345259 


•5SS145 


6 


•35&650 


•554550 


: 1 


•332384 


•521640 


7 


•345479 


•538422 ' 


7 


•358774 


•554821 


8 


•3:>2600 


•521923 


8 


•345699 


•5:58698 ' 


8 


•858£97 


•555(.91 


9 


•332S17 


•522206 


9 


•845919 


-535975 


9 


•859220 


-555861 


10 


•333034 


•5224S9 


10 


•346139 


-5S9251 


10 


•859443 


-555631 


11 


•3^33251 


•522771 


11 


■:340359 


•539527 


' 11 


•869666 


•555900 


12 


•333468 


•523054 


12 


•346580 


-5£9803 


i 12 


•3c9890 


•556170 


13 


•333684 


•52:3336 


13 


•346800 


■540079 


13 


•860114 


•556440 


14 


•3:33901 


•523618 


14 


•347020 


•54C354 


14 


•36(837 


-5567(9 


15 


•3:34118 


•5*23900 


15 


•347-240 


•540630 


15 


•86(561 


•55C979 


16 


•3:34:335 


•5241S2 


16 


•347461 


•54(906 


16 


•36C'784 


•567248 


i 1^ 


•334552 


•524464 


17 


•347681 


•541181 


17 


•3610C8 


-667517 


i 18 


•3:^770 


•524746 


18 


•347901 


•541456 


18 


•361232 


•567786 


; 19 


•334987 


•525C28 


19 


•34S122 


•541731 


19 


•361466 


-558C55 


: 20 


•3:35204 


•525309 


20 


•348:342 


•542006 


20 


•361680 


•555524 


' 21 


•3:35421 


•525591 . 


21 


•348563 


•542281 


21 


•861904 


•558593 


1 22 


•3:35638 


•5-25872 


22 


•34S784 


•542556 


22 


•362128 


•5^8862 


! 23 


•3:35556 


•526153 


23 


•349005 


•542831 


23 


•862352 


•5c9181 


i 24 


•336073 


•526434 


24 


•349-226 


•543106 


24 


•362576 


-r593£9 


• 25 


•336291 


•526715 


25 


•349447 


-543381 


25 


•362800 


-559668 


i 26 


•3365{J9 


•526997 


26 


•349668 


-543656 


26 


•368(24 


-5c 9936 


! 2T 


•336727 


•527278 


27 


•849SS9 


•543930 


27 


•368249 


-56(204 


1 28 


•336944 


•527558 


28 


•350110 


-544204 


, 28 


•363473 


-560472 


' 29 


•3:37162 


•527838 


29 


•35(331 


•544479 


i 29 


•863697 


•560740 


; 30 


•337380 


•528119 


30 


•350552 


-544753 


30 


•363922 


•561CC8 


i 81 


•337598 


•528400 


31 


•350773 


-545G26 


: 31 


•864146 


-561276 


I 32 


•337816 


•528680 


82 


•35C994 


-545300 


82 


•364371 


•561544 


33 


•338034 


•52S960 


33 


•351215 


-545574 


33 


•864595 


•561811 


34 


•33S252 


•529240 


84 


•351437 


•545548 


34 


•364820 


•562079 


35 


•33S470 


•529520 


35 


•351659 


•546122 


85 


•865045 


•562846 


; 36 


•838688 


•529S0O 


36 


•351880 


-546S95 


36 


•365269 


•562613 


1 37 


•33S906 


•530080 


37 


•352102 


•546668 


1 37 


•365495 


-562881 


38 


•339124 


•530359 


38 


•852323 


•546941 


38 


•365719 


•563148 


39 


•339342 


•5306:38 


39 


352544 


•547214 


39 


•86c944 


•563415 


40 


•3:39560 


•5:3(918 


40 


•352766 


•547457 


40 


•866169 


•563682 


41 


•339779 


•531197 


41 


•3529S8 


•547760 


41 


•366894 


•568946 


42 


•339998 


-531476 


42 


•353210 


•548033 


42 


•366619 


•564215 


43 


•340216 


•531755 


43 


•353432 


•54S3C6 


43 


•866844 


•564482 


44 


•340435 


-532034 


44 


•353654 


•545579 


44 


•367069 


•564748 


45 


•340654 


•53-2313 


45 


•353876 


•545851 


45 


•867294 


•665C16 


46 


•&40873 


•532592 


46 


•354095 


•5491-24 


46 


-367520 


•565281 


47 


•341C92 


•5:32871 


47 


•354320 


•549:396 


47 


•367745 


•66654T 


48 


•341:311 


•5:33150 


48 


•354542 


•549668 


48 


•867970 


•566813 


49 


•341529 


•533428 


49 


•354764 


•549940 


49 


•368196 


•666079 


50 


•841748 


•533706 


50 


•354987 


-55(212 


60 


•868421 


•666346 


; 51 


•341967 


•533985 


51 


•3552C9 


•550484 


51 


•368647 


•666611 


52 


•342187 


•534263 


52 


•355431 


-550756 


62 


■368873 


-5668T7 


53 


•342406 


•5:34541 


53 


•355653 


-551027 


63 


•369C98 


•567142 


54 


•:342625 


•534819 


54 


•355576 


-551299 


54 


•869324 


•667408 


55 


•342844 


•5:35(97 


55 


•356(98 


•551570 : 


55 


•369550 


•667678 


56 


•343063 


•535074 


56 


•356:321 


•551842 


56 


•369776 


•567939 


57 


•843283 


•535652 


57 


•356544 


•552114 


57 


•370002 


•568204 


, 58 


•34:3502 


•5359-29 


58 


•356767 


•552385 ! 


58 


•370228 


•668469 


59 


•343721 


•536206 


59 


•356990 


•552656 i 


69 


•3704^ 


•568784 


60 


•343941 


•536*84 


. ^' 


•357213 


-5529-27 1 


60 


•370680 


•668999 



External Secants. 



449 



48 DEGEEES. 


49 DEGPwEES. 


50 DEGREES. 


'[ 


Min. 


1 Naf. No. 


1 Logarihm 


Min 




i Nnt No 


1 I..°ati.i..n 


: Min. 


1 Nat No 


1 L..gari:l=n. 


1 





0-494477 


9^c91146 


, 0^524253 


: 9-719541 


1 


0-555724 


9-744859 




1 


•494980 


•691570 


1 


i -524763 


1 -719963 


1 1 


i -556263 


•745280 




2 


•495443 


•694994 


2 


'' -525274 


; •720386 


2 


-556804 


•745702 


I 


8 


•495927 


•695418 


3 


1 -525785 


I -72 '.808 


3 


-557344 


•746123 


1 


4 


•49J412 


•695342 


4 


! -526297 


; ^721231 


4 


•557885 


•746545 


1 


5 


•49j393 


•698266 


5 


•526309 I ^721653 


5 


•553427 


•746966 


1 


6 


•497381 


•696689 


6 


•527322 


•722076 


6 


•558969 


•747333 


j 


T 


•497887 


•697113 


7 


-527835 


•722498 


7 


-559511 


•7478 9 




8 


•493353 


•697537 


8 


-523348 


•722920 


8 


•560054 


•743230 


1 


9 


•498 S40 


•697961 


9 


-528363 


■723343 


9 


•560598 


•748652 


' 


10 


•499327 


•698335 


10 


-529378 


-723766 


10 


•561142 


•749073 




11 


•499314 


•693808 


11 


-529393 


•724188 


11 


•561687 


•749491 




12 


•530302 


•699232 


12 


-530408 


-724610 


12 


•562232 


•749916 




13 


•500790 


•699356 


13 


-53C924 


•725032 


13 


•562778 


•750837 




14 


•501279 


•700079 


14 


-531440 


•725454 


14 


•563824 


-750753 


1 


15 


•501708 


•700503 


1 15 


•531957 


•725877 


15 


•568371 


•751180 


! 


16 


•502258 


•700927 


i 1^ 


•532175 


•726299 


16 


•564418 


•751601 


i 


17 


•512749 


•701351 


1 17 


-532992 


•726721 t 17 


•564966 


•752022 


i 


18 


•5032^9 


•701774 


i 18 


-533510 


•727143 


18 


•565514 


•752443 




19 


•503730 


•702198 


19 


•534029 


•727565 


19 


•566!)63 


•752865 




20 


•5)4221 


•702621 


20 


•534548 


•727937 


20 


•566612 


•753286 




21 


•504713 


•703045 


21 


•535068 


•7284C9 


21 


•5671C2 


•758707 




22 


•5)5205 


•703463 


22 


•535539 


•723S32 


22 


•567712 


•754128 




23 


•5)5G9S 


•703391 


23 


•536110 


•7^29254 


23 


•568263 


•754549 




24 


•506191 


•704315 


24 


•536631 


•729676 


24 


•568815 


•754971 




25 


•50G'J35 


•704738 


25 


•537153 


•730C93 


25 


•569867 


•755392 




26 


•507130 


•705162 


26 


•537675 


•730520 


26 


•569919 


•755813 




27 


•507074 


•705585 


27 


•538193 


•730942 


27 


•570472 


•756214 




23 


•508169 


•706008 


23 


•538721 


•731364 


23 


•571025 


•756655 




29 


•508j64 


•706431 


29 


•539245 


•731786 1 


29 


•571579 


•'757076 




81) 


-509160 


•706854 


30 


•539769 


•732208 


30 


•572184 


•757493 




81 


•509357 


•707278 


31 


•540294 


•732630 I 


31 


•572689 


•757919 




82 


•510154 


■707701 


32 


•540819 


•733C52 ! 


82 


•573244 


•753340 




83 


•51.0651 


•708124 


33 


•541345 


•738474 ; 


3S 


•573300 


-753761 




84 


•511148 


•708547 


34 


•541871 


•783896 i 


34 


•574357 


•7r9182 




85 


•511646 


•708970 


35 


•542398 


•734318 • 


05 


•574914 


•7c9303 




86 


•512145 


•709394 


36 


•542925 


•734740 1 


36 


•575472 


■760024 




87 


•512645 


•709317 


37 


•543452 


•735161 ' 


87 


•576030 


-760445 




88 


•513145 


•710240 


38 


•543930 


•735583 : 


38 


•576589 


•760866 




89 


•513045 


•710663 


39 


•544508 


•786004 i 


89 


•577148 


•761287 




40 


•514146 


•711087 


40 


•545087 


•736426 1 


40 


•5777v8 


•761708 




41 


•514G47 


•711509 


41 


-545567 


•736848 ; 


41 


•578268 


•762129 




42 


•515148 


•711932 


42 


-546097 


•737270 ; 


42 


'578829 


-762550 




43 


•515650 


•712355 


43 


•546628 


•737692 


43 


•579390 


•762971 




44 


•516152 


•712778 


44 


-547159 


•738114 j 


44 


•579952 


-763892 




45 


•516655 


•713200 


45 


•547090 


•733535 I 


45 


•580514 


-763813 




46 


•517158 


•713623 


46 


•548222 


•738957 


46 


•581077 


•764234 




47 


•517662 


•714046 


47 


•548755 


•739379 


47 


•581641 


•764655 




48 


•518166 


•714469 


48 


•549288 


•739800 


48 


•582205 


•765076 




49 


•518670 


•714892 


49 


•549321 


•740221 


49 


•582770 


•765497 




50 


•519175 


•715314 


50 


•550355 


•740643 


50 


•583335 


765918 




51 


•519!)81 


•715737 


51 


•550890 


•741065 


51 


•583900 


•766839 




52 


•520188 


•716160 


52 


•551426 


•741487 


52 


•534466 


•766760 




53 


•520695 


•716583 


53 


•551961 


•741908 


53 


•585033 


•767181 




54 


•521202 


•717006 


54 


•552497 


•742330 : 


54 1 


•5S5600 


-767602 




55 


•521709 1 


•717428 1 


55 


•553033 


•742751 


55 , 


•586168 


•768022 




56 


•522216 


•717850 


56 


•558571 


•743173 i 


56 


•586787 


•76844:3 




57 


•522725 


•718273 


57 


•554109 


•743595 


57 


•587306 


•768864 




53 


'523234 


•718696 


58 


•554647 


•744017 


58 ! 


•587875 


•769285 




59 


•523740 


•719118 


59 


•555185 


•744438 I 


59 


•588445 


•769706 




60 


•524253 


•719541 


60 


•555724 


•744859 


60 


•589016 


•770127 

1 












9ft* ~ 










1 



450 








Versed 


Sines. 








51 DEGREES. 






LEES. 


|! 


53 DEGPwEES. 




52 DEGP 


Min 


' Nat. No. 


Logarithm 


Min 


Nat No 


1 Lopariihm. 


|i Min. 


1 Nat No. 


Logarithm. 





0-3706S0 


9-56S999 


'! 


0^3S4389 


, 9-584714 





0-398185 


-600085 


1 


1 -87(906 


•569264 


i 1 


: •3S456S 


1 -584973 


1 


•398417 


-60C338 ; 


1 


1 •371182 


•569528 


11 2 


•884797 


•585232 


2 


•398650 


-60C591 


3 


i •371358 


•569793 


i! 3 
■I 4 


1 -385027 


•585491 


3 


•398882 


•600845 ■ 


4 


j -871584 


•570057 


' -385256 


-585749 


4 


•S99115 


•601098 


5 


1 -871810 


: •570322 


i 5 


' -385485 


•586008 


5 


•S99347 


•601351 


6 


1 •87-2037 


i -570586 


1 ^ 


1 -885715 


•586266 


^ 6 


•399580 


•601603 


7 


j -872263 


•570850 


1 7 


; -385944 


•586525 


7 


•399812 


•601856 ^ 


8 


, -872490 


: -571114 


i 8 


I •386174 


•586783 


8 


•400045 


•602109 


9 


•872716 


•571378 


9 


: -386404 


•5S7041 


9 


•400278 


•602362 


10 


' •872943 


■ -571642 


10 


j •386633 


•587299 


10 


•400510 


•602614 


11 


•373170 


■ -571906 


11 


-386863 


•587557 


11 


•400743 


•602867 


12 


•873396 


•572170 


12 


: -387093 


1 ^587815 


12 


•40(976 


•603119 


13 


•378623 


•572434 


13 


: •387823 


i '588073 


13 


-401209 


•603371 


14 


•373850 


•572697 


14 


i •387553 


1 •58S331 


14 


•401442 


•603623 


15 


•874076 


•572960 


15 


•8S7783 


•588588 


15 


•401675 


•603875 


16 


•374303 


•5732-24 


16 


•888013 


•588846 


16 


•401908 


•604127 


17 


•374530 


•5784S7 


17 


•388243 


•589103 


17 


•402142 


•604379 


18 


•374757 


•573750 


18 


•388473 


-589861 


18 


•4C2375 


-604631 


19 


•374984 


•574013 


19 


•3SS703 


•589618 


19 


-402608 


•604S82 


20 


•875211 


•574276 


20 


•388933 


•589S75 


20 


-402841 


•605134 


21 


•375489 


•574539 


! 21 


•3891 64 


-590132 


21 


•403074 


-605355 


22 


•875666 


-574802 


22 


•3S9394 


-590389 


22 


•403808 


•605637 


23 


•375893 


•575064 


23 


•889624 


•590646 


23 


-403541 


-605888 


24 


•376120 


•575327 


24 


•389855 


-590903 


24 


•403775 


-606139 


25 


•876848 


•575589 


25 


-890085 


•591160 


25 


•4040(9 


•606891 


26 


•376575 


•575S52 


26 


-390316 


-591416 


26 


-404242 


•606642 


27 


•376803 


•576114 


27 


•390547 


•591673 


27 


•404476 


-606893 


28 


•377030 


•576876 


28 


•390777 


•5919-29 


28 


•404710 


-607144 


29 


•877258 


•576638 


29 


•391008 


•592186 


29 


•404943 


•607894 


30 


•377485 


•576900 


30 


•391239 


•592442 


30 


•405177 


•607645 


31 


•877718 


•577162 


31 


•391469 


•592698 


31 


•405410 


-607896 


32 


•877941 


•577424 


32 


•391700 


•592954 


32 


•405645 


•608146 


33 


•87S168 


•577685 


33 


•391931 


•593210 


33 


•405879 


-608397 


84 


•87S396 


•577947 


34 


•892162 


•593466 


34 


•406113 


•608647 


35 


•87S624 


•578208 


35 


•392393 


-593721 


35 


-406347 


•608897 


36 


•37SS52 


•57S470 


36 


•392624 


-598977 


36 


•406581 


•609147 


37 


•879080 


•57S731 


37 


•892855 


•594233 


37 


•406815 


-609397 


38 


■87930S 


•57S992 


38 


•393086 


-594488 


38 


•407049 


•609647 


39 


•379.586 


•579-253 


39 


•398817 


•594748 


39 


•407284 


■609897 


40 


-879764 


•579514 


40 


•393549 


•594999 


40 


•407518 


•610147 


41 


•879992 


•579775 


41 


•393780 


•595254 


41 


•407753 


•610397 


42 


•880221 


•580086 


42 


•394012 


•595509 


42 


•407987 


•610646 


43 


•380449 


•580297 


43 


•894243 


•595764 


43 


•408221 


-610896 


44 


•380677 


•580557 


44 


•394474 


•596019 


44 


•4C8456 


•611146 


45 


•3SG906 


•580818 


45 


•394706 


•596274 


45 


•408690 1 


•611394 


46 


•3S1134 


•581078 


46 


-394938 


•596528 


46 


•408925 


-611644 


47 


•381863 


•581339 


47 


•395169 


•596783 ! 


47 


•409160 


-611893 


48 


•881592 


•581599 


48 


•395401 


•597038 . 


48 


•409394 


•612142 


49 


•8818-20 


•581859 


49 


•395632 


•597292 


49 


•409629 


•612891 


50 


•382049 


•5S2119 


50 


•895864 


•597546 ; 


50 


•409864 


•612640 


51 


•882278 


•582379 


51 


•396096 


-597801 ' 


51 


•410099 


•612S88 


62 


•3S2506 


•582639 


52 


-896828 


-598055 ; 


52 


•410334 


•613137 


53 


•8S2785 


•5S2S99 


53 


-396560 


•598309 1 


53 


•410569 


•613386 


54 


•8S29G4 


•5S315S 


54 


•396792 


•598563 ; 


54 


•410804 


•618634 


55 


•883193 


•583418 


55 


•897024 


•598S17 : 


55 


•411039 


•613883 


56 


•3S3422 


•583677 


56 


•397256 


•599071 


56 


•411274 


•614131 


57 


•383657 


•583937 


57 


•397488 


-599324 


57 


•411509 


-614379 


58 


•3S3SS0 


•584196 : 


58 


•897720 


-599578 ; 


58 


-411744 


•614627 


59 


•384110 


•584455 


59 


•397953 


•599832 1 


59 


•411979 


•614875 


60 


•384339 


•584714 i 

1 


60 


•398155 


-600085 j 


60 


•412215 


•615124 



External Secants. 



4)1 





51 DEGREES. 


T 


52 DEGREES. 


1 53 DEGREES. 


1 Min 


Nat. No. 


Logarithm 


i Min 


1 Nat No 


! I.ogaritlim 


i Mm 


I Nif No 


1 Logariibju , 


1 


0-589016 


9-770127 


! 


0^624269 


j 9-795372 





0-661640 


9^S20622 


1 


•589587 


•770548 


1 


•624875 


•795793 


1 


•662282 


•821043 


2 


•590159 


•770969 


2 


•625480 


•796214 


2 


•662924 


•S21464 


i s 


•590731 


•771389 


3 


•626C86 


1 -796634 


3 


•663567 


•821 885 


4 


•591803 


•771810 


4 


•626693 


1 •797065 


4 


•664211 


•822806 


5 


•591876 


•772231 


5 


•627300 


1 ^797476 


5 


•664855 


-822727 \ 


6 


•592450 


•772652 


6 


•627908 


•797896 


6 


•665500 


-S28148 1 


7 


•593025 


•773073 


7 


•628517 


•798317 


7 


666145 


•8285C9 


8 


•593600 


•773494 


8 


•629126 


•798738 


8 


•666791 


•823990 


9 


•594175 


•773914 


9 


•6^.9736 


•799158 


9 


•667489 


•824411 


10 


•594751 


•774335 


10 


•630346 


•799579 


10 


•668086 


•824883 


11 


•595327 


•774756 


11 


•630957 


•800000 


11 


•668734 


•825254 


12 


•595904 


•775177 


12 


•631569 


•800421 


12 


•669383 


•825675 


i 13 


•596482 


•775698 


13 


•6321S1 


•80C841 


1 13 


•670032 


•826(96 


1 ^^ 


•597060 


'776018 


14 


•632794 


•801262 


1 14 


•670682 


•S26517 


1 15 


•597639 


•776439 


15 


•633407 


•801683 


: 15 


•671333 


•820988 


i ^6 


•598219 


•776860 


16 


•634C21 


•802104 , 


i 16 
17 


•671985 


•827860 


1 IT 


•698799 


•777281 


17 


•634635 


•802524 ^ 


•672637 


•827781 


! 18 


•599380 


•777702 


18 


•635251 


•802945 


j 18 


•678290 


•8282(2 


19 


•699960 


•778122 


19 


•635867 


•803366 


19 


•678943 


■828623 


20 


•600642 


•778543 


20 


•636483 


•803787 


1 20 


•674597 


•829044 


21 


•601124 


•778964 


21 


•637100 


•804207 


1 21 


•675252 


•829466 


22 


•601706 


•779385 


22 


•637717 


•804628 


22 


•676907 


•829887 


23 


•602289 


•779805 


23 


•638335 


•805049 


23 


•676563 


•88(308 


24 


•602873 


•780226 


24 


•638954 


•805470 


24 


•677220 


•830729 


25 


•603458 


•780647 


25 


•639574 


•805891 


25 


-677877 


•881161 


26 


•604043 


•781068 


26 


•640194 


•806311 


26 


•678534 


•831572 


2T 


•604628 


•781488 1 


27 


•640814 


•806732 


27 


•679193 


•831993 


28 


'605214 


•781909 i 


28 


•641435 


•807153 


28 


•679852 


•882415 


29 


•605800 


•782330 i 


29 


•642057 


•807574 


29 


•C80512 


•882886 


30 


•6C6387 


•782750 


30 


•642680 


•807995 


30 


•681173 


•888257 


31 


•606975 


•783171 ! 


31 


•643303 


•808415 


31 


•081834 


•888679 


32 


•607564 


•783592 i 


32 


•643926 


•80S836 


82 


•682496 


•884100 


83 


•608153 


•784013 ' 


33 


•644550 


•8C9257 


33 


•688159 


•884622 


34 


•608742 


•784433 1 


34 


•645175 


•809678 ; 


34 


•688822 


•834943 


35 


•609332 


•784854 ; 


35 


•646801 


•810(99 1 


35 


•684486 


•886864 


36 


•609923 


•785275 i 


36 


•646427 


•810520 ! 


36 


•685150 


•885786 


3T 


•610514 


•785696 


37 


•647054 


•810940 


37 


•685815 


•886207 


38 


•611106 


•786116 


38 


•647681 


•811361 i 


38 


•686481 


•886629 


39 


•611698 


•786537 


39 


•648309 


•811782 1 


89 


•687148 


•837050 


40 


•612291 


•786958 


40 


•648938 


•812203 i 


40 


•C87815 


•837472 1 


41 


•612884 


•787378 


41 


•649567 


•812624 ! 


41 


-688483 


•887893 ' 


42 


•613478 


•787799 


42 


•650197 


•813045 ! 


42 


•689162 


•888316 1 


43 


•614073 


•788220 


43 


•650827 


•813466 j 


43 


-689821 


•888736 i 


44 


•614668 


•788640 


44 


•651458 


•813887 


44 


•690491 


•889158 I 


45 


•615264 


•789061 


45 


•652090 


•814307 


45 


•691161 


•889579 


46 


•615860 


•789482 


46 


•652722 


•814728 


46 


•691882 


•840001 


47 


•615457 


•789903 


47 


•653855 


•815149 


47 


•692504 


■84C423 1 


48 


•617054 


•790323 


48 


•653989 


•815570 i 


48 


-693177 


•840844 ' 


49 


•617652 


•790744 


49 


•654623 


•815991 1 


49 


•698850 


•841266 i 


50 1 


•618251 


•791165 


50 


•655258 


•816412 


50 


•694524 


•841688 


51 1 


•618850 


•791586 


51 


•655893 


•816833 


51 


•C95199 


8421C9 I 


52 


•619450 


•792006 


52 


•656529 


•817254 ! 


52 


•096874 


•842531 j 


53 


•620050 


•792427 


53 


•657166 


•817675 : 


58 


•096550 


842953 


54 


•620651 


•792848 


54 


•657803 


•818096 : 


54 


•697227 


•848874 1 


55 


•621253 


•793268 


55 


•658441 


•818517 


55 


-697904 


•843796 j 


56 


•621855 


•793689 


56 


•659080 


-818938 1 


56 


•698582 


•844218 


57 


•022458 


•794110 


57 


•659719 


•819369 


57 


•699261 


•844689 ! 


58 


•623(!61 


•794581 


58 


•660359 \ 


•819780 


58 


•699941 i 


•845061 i 


59 


•623665 


•794951 : 


59 


•660999 


•820201 


59 


•700621 1 


•S45483 1 


60 


•624269 


•795872 j 


60 


•661640 1 

1 


•820622 j 


60 


•701302 : 

1 


•846005 ' 



452 



Versed Sines. 





54 DEGREES. 




55 DEGE 


EES. 




56 DEGE 


EES. 










Min 


1 Nat No 


1 Loj;ariilim. 


Mill 


1 Nat. Nc- 


Logarithm 


!Min. 


1 Nat No 


Logarilhm. 





0-412215 


i 9-615124 





j 0-426423 


9-629841 





0^44(:807 


9-644249 


1 


•412450 


1 -615371 


1 


•426662 


•630C84 


1 


•441048 


•644486 


2 


1 -412685 


•615619 


2 


•426900 


-630826 


2 


•441289 


•644724 


3 


1 412921 


•615867 


3 


•427139 


•630569 


3 


•441531 


•644961 


4 


i -413156 


•616114 


4 


•427877 


•680811 


4 


•441772 


•645198 


5 


i -418392 


•616362 


i S 


•427616 


•681054 


5 


•442013 


•645435 


6 


' -413628 


1 •616610 


« 


•427854 


•681296 


6 


•442255 


•645673 


7 


-418863 


j -616857 


i 7 


•428098 


•631588 


7 


•442496 


-645910 


8 


•414099 


•617104 


8 


•428331 


•081780 


8 


•442738 


-646147 


9 


-414835 


•617351 


9 


•428570 


•632022 


9 


•442980 


-646384 


10 


-414571 


-617599 


10 


•428809 


•682264 


10 


•443221 


-646620 


11 


414807 


•617846 


11 


•429047 


•632505 


11 


•443463 


•646857 


12 


-415042 


-618092 


12 


•429286 


•632747 


12 


•443704 


-647C94 


13 


415278 


•018889 


13 


•429525 


•682989 


13 


•443946 


-647330 


14 


-415514 


•61S586 


14 


•429764 


•638280 


14 


•444188 


•647567 


15 


415750 


•618838 : 


15 


480003 


•688472 


1 15 


•444430 


•647803 


16 


-415986 


•619079 ; 
•619323 ' 


16 


480042 


•638713 


1 16 


•444672 


-648C40 


17 


-416223 


17 


•430481 


•683954 


17 


•444914 


-648276 


18 


•416459 


•619572 ' 


18 


•480720 


•684195 


18 


•445156 


•648512 


19 


•416695 


•619818 i 


19 


•480960 


•684437 


19 


•445398 


•648748 


20 


•416981 


•620065 j 


2) 


•431199 


•684678 


20 


•445640 


•648984 


21 


•417168 


•620311 ; 


21 


481438 


•634919 


21 


•445852 


•649220 


22 


-417404 


•620557 ! 


22 


•481677 


•685159 


22 


•446224 


•649456 


23 


•417641 


•620808 i 


28 


•431917 


•635400 


23 


•446866 


•649691 


24 


•417877 


•621049 


24 


•482156 


•635641 


24 


•446608 


•649927 


25 


•418114 


•621294 ' 


25 


•482896 


•635881 


25 


•446851 


•650162 


26 


•418350 


•621540 ' 


26 


•482685 


•686122 


26 


•447098 


•650398 


27 


418587 


•621786 ' 


27 


•432875 


•636362 


27 


•447335 


•650638 


28 


•418823 


•622031 . 


28 


•483114 


•686608 


28 


•44757S 


•650869 


29 


•419060 


•622276 i 


29 


•488354 1 


•6:36843 j 


29 


•447820 


•651104 


30 


•419297 


•622522 : 


30 


•438594 


•637083 1 


80 


•448068 


•651389 


31 


•419534 


•622767 i 


31 


•488888 


•687823 


31 


•448306 


•651574 


32 


•419771 


•623012 1 


32 


434073 


•687568 


32 


•448548 


•6518C9 


33 


•420008 


•628257 i 


33 


•434813 


•637808 


83 


•448791 


•652044 


34 


•420245 


•628502 i 


34 


•484558 


•638043 1 


34 


•449C34 


•652279 


35 


•420482 


•628747 I 


35 


^484793 


•6382S3 1 


35 


•449276 


•652514 


36 


•42a719 


•623992 1 


36 


•485033 


•638522 


36 


•449519 


-652748 


37 


•420956 


•624-237 i 


37 


•485278 


•638762 1 


37 


•449762 


-652983 


38 


•4-21193 


•6-24481 ! 


38 


•435513 


•689001 


38 


•450005 


•653217 


89 


•421480 


•624726 I 


39 


•485753 


•689241 


89 


•450248 


•658452 


40 


•421668 


•624970 1 


40 


•435993 


•689480 


40 


•450491 


•653686 


41 


•421905 


•625215 1 


41 


-436284 


•639719 


41 


•450784 


•653920 


42 


•422143 


•625459 


42 


-486474 


•689958 


42 


•45C977 


•654155 


43 


•422880 


•625703 


43 


-486714 


•640197 


43 


•451220 


•654389 


44 


•422617 


•625947 ! 


44 


•486955 


•640486 


44 


•451463 


-654623 


45 


•422855 


•626191 


45 


•487195 


•640675 i 


45 


•451707 


•654857 


46 


•428092 


•626435 ; 


46 


•487435 


•640914 


46 


•451950 


-655090 


47 


•428880 


•626679 I 


47 


•487676 


•641153 


47 


•452198 


•655324 


48 


-428568 


•626928 ' 


48 


•487916 


•641891 


48 


•452487 


-655558 


49 


•423805 


•627166 


49 ! 


•438157 i 


•641680 1 


49 


•452680 


-655791 


50 


•424043 


•627410 1 


50 


•438898 ' 


•641868 


50 


•452924 


•656025 


51 


'424281 


•627654 1 


51 


•488689 


•642107 


51 


•458167 


•656258 


52 


•424519 


•627897 ' 


52 


•488879 ' 


•642345 


52 


•458411 


•656492 


53 


•424757 


•628140 i 


58 ! 


•439120 


•642583 


58 


•458654 


•656725 


54 


•424995 


•628384 ' 


54 i 


•489361 1 


•642821 1 


54 


•458898 


•65695S 


55 


•425288 


•628627 


55 


•489602 1 


•643060 i 


55 


•454142 


•657191 


56 


•425471 


•628870 : 


56 i 


•439843 


•643298 ! 


56 


•454385 


•657424 


57 


•425709 


•629113 ^ 


57 ! 


•440084 j 


•648535 i 


57 


•454629 


•657657 


58 


•425947 


■629856 1 


58 1 


•440325 i 


•648773 1 


58 


•454878 


•657890 


59 


•426185 


•629598 


59 


•440566 i 


•644011 


59 


•455117 


•658123 


60 


•426423 


•629841 


60 


•440807 


•644249 


60 


•455361 


•658356 



External Secants. 



64 BEGEEES. 


55 DEGREES. 


56 BEGEEES. 


Mill. 


Nat No. 


j Logariilim 


1 Min. 


1 Nat. No. 


Loearidim. 


Min. 


1 Nat. No. 


1 L();;arithm. 





1 0-701302 


9-845905 





! 0-743447 


i 9-871250 





1 0-788291 


1 9896687 


1 


! -701983 


•846827 


1 


! .744172 


i ^871673 


1 


1 •789063 


1 ^897112 


2 


! -702665 


•846749 


2 


•744897 


i -872096 


2 


•789836 


i •897537. 


3 


1 -703848 


•847170 


3 


•745623 


1 -872519 


3 


•790609 


1 ^897962 


4 


1 -704032 


•847592 


4 


•746350 


i ^872942 


4 


•791383 


•898387 


5 


-704716 


•848014 


5 


•747078 


: -878366 


5 


•792158 


•S9S812 


6 


•705401 


•848436 


6 


•747806 


1 •873789 


6 


•792984 


•899237 


T 


-706087 


•848858 


7 


•748585 


! ^874212 


7 


-793710 


•899662 


8 


•706773 


•849280 


8 


•749265 


! ^874635 


8 


•794483 


•900087 


9 


•707460 


•849702 


9 


•749996 


1 ^875059 


9 


•795266 


•90C512 


10 


•708148 


•850124 


10 


•750727 


1 ^875482 


10 


•796045 


•900938 


11 


•708836 


•850546 


! 11 


•751459 


! ^875905 


11 


•796825 


•901363 


12 


-709525 


•850968 


12 


•752192 


! •876329 


12 


•797606 


•901788 


13 


•710215 


•851390 


13 


•752926 


1 ^876752 


13 


•798387 


•902213 


14 


-71(906 


•851812 


14 


•753661 


! ^877176 


14 


•799169 


•902639 


15 


-711597 


•852234 


15 


•754896 


1 -877599 


15 


•799952 


•908064 


16 


1 -712289 


•852656 


16 


•755132 


•878028 


16 


•800737 


•90349 > 


17 


•712982 


•853078 


17 


•755869 


•878447 ^ 


17 


•801521 


•903015 


18 


•713675 


•853500 


18 


•75C6C6 


•878870 , 


18 


•802307 


-904341 


19 


•714369 


•853923 


19 


•757345 


1 -879294 , 


19 


•803094 


•904766 


20 


•715064 


•854345 


20 


•758084 


i -879717 


20 


•803881 


-905192 : 


21 


•715760 


•854767 


21 


•758824 


i -880141 


21 


•804669 


-905617 


22 


•716456 


•855189 


22 


•759564 


i -880564 


22 


•805458 


•906043 : 


23 


•717153 


•855612 


23 


•760305 


•88C988 


23 


•806248 


•9064G9 


24 


•717850 


•856034 


24 


•761048 


•8S1412 ; 


24 


•807089 


-906S94 


25 


•718548 


•856456 


25 


•761791 


•881836 


25 


•807830 


-907823 


26 


•719247 


•856878 


26 


•762535 


-882260 i 


26 


•808623 


•907746 


2T 


•719947 


•857301 


27 


•763279 


•882683 ': 


27 


•809416 


-908172 


28 


•720648 


•857723 


28 


•764024 


•883107 ! 


28 


•810210 


•908598 


29 


•721349 


•858145 


29 


•764770 


•883531 i 


29 


•811005 


•9C9024 


30 


•722051 


•858568 


30 


•765517 


•883955 i 


30 


•811801 


•9:9450 


31 


•722753 


•858990 


31 


•766265 


•884379 ! 


31 


•812598 


-909876 


32 


•728457 


•859412 


82 


•767013 


•884803 ! 


82 


•813395 


•910302 i 


83 


•724161 


•859885 


33 


•767762 


•885227 1 


33 


•814193 


•9107'28 


84 


•724866 


•860257 


34 


•768512 


•885651 I 


84 


•814998 


•911154 i 


35 


•725571 


•860680 


85 


•769263 


•886075 i 


85 


•815798 


•911580 1 


86 


•726277 


•861103 


36 


•770014 


•886499 ! 


36 


•816594 


•912006 ' 


87 


•72G984 


.861525 


37 


•770767 


•886923 ; 


37 


•817396 


•912432 ; 


38 


•727692 


•861947 


88 


•771520 


•887847 i 


38 


•818199 


•912859 i 


39 


•7284111 


•862370 


39 


•772274 


•887712 : 


89 


•819002 


•9132S5 ! 


40 


•729110 


•862793 


40 


•778029 


•888196 i 


40 


•819806 


•913711 


41 


•729320 


•863216 


41 


•773784 


-888620 : 


41 


•820611 


•914138 


42 


•730530 


•8636S8 


42 


•774540 


•8S9044 


42 


•821418 


•914564 


43 


•731241 


•864061 


i 43 


•775298 


-889469 i 


43 


•822225 


•914991 ; 


44 


•731953 


•8644S4 


44 


•776056 


•8S9S93 1 


44 


•823033 


•915417 


45 


•732666 


•864906 


1 45 


•776815 


•890817 


45 


•823842 


•915846 ^ 


46 


•733380 


•865329 


1 46 


•777574 


-89:)742 ' 


46 


•824651 


-916270 i 


47 


•734C94 


•805752 


47 


•778334 


•891166 1 


47 


•825462 


•916697 i 


48 


•7348(;9 


•866175 


48 


•779095 


•891591 


48 


•826273 


•917124 


49 


•735525 


•866597 


! 49 


•779S57 


•892015 ! 


49 


•827085 


•917550 


50 


•736241 


•867020 


i 50 


•780620 


•892440 ^ 


50 


•827898 


•917977 : 


51 


•786958 


•867443 


i 51 


•781384 ! 


•892864 


61 


•828712 


-9184C4 


52 


•737676 


•867866 


1 52 


•7S2148 


•8982S9 I 


52 


•829527 


•9188;^1 


53 


•738395 


•8682S9 


53 


•782913 1 


•893714 


58 


•830843 


•919258 


54 


•739115 


•868712 


54 


•783679 


•894138 


54 


•831160 


•919685 


55 


•739835 


•869135 


1 55 


•784446 


-894563 


65 


•831977 


-921.112 , 


56 


•740556 


•869558 


56 


•78,5213 1 


•894988 


56 


-832796 


•920539 , 


57 


•741277 


•869981 


57 


•7859S1 


-895412 


57 


•833615 


•92(966 


58 


•742000 


•870404 


58 


•786750 


•895837 ! 


58 


•834435 


•921393 ; 


59 


•742728 


•870827 


59 


•787520 


•896262 i 


59 


•835256 


•921820 ! 


eo 


•748447 


•871250 


i '' 


•788291 


•S966S7 i 


60 


•836078 1 


•92-2247 1 



454 










Versed 


Sines. 
















57 DEGEEES. 


58 DEGREES. 


i 


59 DEGEEES. 


M,„. 


i Nat No 


( I.oKaiiihm. ij J\!ii:. 


, Nat. No. 


i Log,ritl:.„ 


Min. 


: N?,t N.. 


I.. -iMilim 


' 


0455361 


9-658356 ;i 


i 0-470081 


9-672173 


, 


I 0484962 


9-685708 


1 


•455605 


•658588 li 1 


•470327 


•672400 


1 


1 -485211 


•f 8c 981 


2 


•455849 


•658821 i 2 


•470574 


•672628 


1 2 


! -485460 


•686154 


S 


•456(;93 


•659054 i 3 


•470821 


•672856 


li 3 


i 485710 


•f8eg77 


' 4 


•456837 


•659286 ! 4 


•471068 


•673183 


^1 4 


•485960 


•68^660 


5 


•456581 


•659518 i 


5 


•471315 


•678311 


:l 5 


•486209 


•086823 


6 


•456825 


•6c9750 1 


6 


•471562 


•673589 


i 6 


•486459 


•687046 


7 


•457070 


•659982 '! 7 


•4718C9 


•678766 


u 7 


1 -486708 


•687269 


8 


•457814 


•66C215 ii 8 


•472t56 


•673998 


li 8 


1 48C9.':8 


•687492 


9 


•457558 


•66C446 ': 9 


•472303 


•674221 


!: 9 


487207 


•687714 


! 10 


•457803 


•660678 i! 10 


•472c50 


•674448 


1 10 


•487457 


•087937 


■ 11 


•458047 


•660910 ,i 11 


•472797 


•674675 


i 11 


•487707 


•6S81£9 


12 


I -458292 


•661142 i 12 


•473 44 


•6749(2 


: 12 


•487957 


•088882 


i 13 


•458536 


•661374 13 


•473291 


•675129 


1 18 


•488207 


1 •688604 


! 14 


•458781 


•661605 i 14 


•473589 


•675856 


1 14 


•488457 


•688S26 


: 15 


•459'^ 25 


•661837 ' 15 


•473786 


•675582 


\ 15 


•488707 


•689048 


• 16 


•459270 


•662068 ! 16 


•474C33 


•675809 


i 16 


•485957 


•689270 


: 17 


•459515 


•662300 ! 17 


474281 


•676036 


! 17 


489207 


•089492 


1 18' 


-459760 


•662531 \\ 18 


•474528 


•676262 


; 18 


4S9457 


•689714 


19 


•460004 


•662702 1 19 


•474776 


•676489 


i 19 


•489707 


•C89986 


20 


•460249 


•662943 :; 20 


475023 


•676715 


' 20 


•489957 


•69G158 


21 


•400494 


•663224 ii 21 


475271 


•676941 


' 21 


•490207 


•69C380 


22 


•460789 


•663455 : 22 


475518 


•677167 


; 22 


•49(:458 


•6906(2 


23 


•460984 


•663686 i 23 


•475766 


•677S94 


j 23 


•490708 


•690823 


24 


•461229 


•663917 ij 24 


•476014 


•677620 


' 24 


490958 


•691045 


25 


•461474 


•664147 ! 25 


•476262 


•677846 


25 


•491 2f 9 


•691266 


26 


•461719 


•664378 ,f 26 


•476510 


•678G72 


1 26 


•491459 


•091488 


27 


•461965 


•664609 ■ 27 


•476758 


•678298 


: ,27 


•491710 


•6917(9 


28 


•462210 


•664839 28 


477C05 


•678523 


28 


•491960 


•691980 


29 


•462455 


•665070 ; 29 


477253 


•678749 


: 29 


•492211 


•692151 


80 


•462700 


•665300 : 30 


•477501 


•678975 


: 30 


•492462 


•6S2872 


31 


•462940 


•665530 '' 31 


477749 


•679200 


1 81 


•492712 


-C02593 


32 


•403191 


•665760 32 


•477997 


•679426 


: 82 


•492933 


•692814 


33 


•463436 


•665990 : 33 


478246 


•679651 


; S3 


•498214 


•093685 


34 


•463682 


•666220 ' 34 


•478494 


•679S76 


! 34 


493465 


•693256 


35 


•468927 


•666450 35 , 


478742 


•68Glf.2 


' 35 


498716 


•093477 


36 


•46-1173 


•666680 : 36 \ 


•478991 


-68CS27 


; 86 


•498966 


-093697 


37 


•464419 


•666910 37 i 


•479219 


•68C552 


37 


•494217 


•098918 


38 


•464664 


•667140 38 ! 


479487 


•08O777 


38 


•494468 


•094188 


39 


•464910 


•667369 1 39 


•479735 


•C81O02 


£9 


•494719 


-094359 


40 


•465156 


•667599 40 


•479984 


•681227 


40 


49.^970 


-094579 


41 


•4654i,2 


•667828 41 


•480232 


•C81451 


41 


•495221 


•6947r;9 


42 


•465648 


•668058 1 42 


•480481 


•681676 


42 


•495472 


•695019 


43 


•465S94 


•668287 J| 43 


•480730 


•681901 


43 


495724 


•695240 


44 


•466140 


•668516 I 44 


•480978 


•682125 


44 


•495975 


•695460 


45 


•466886 


•668745 : 45 


•481227 


-682350 


45 


•496226 


-695680 


46 


•466682 


•668974 46 


•481475 


•682574 


46 


•496477 


•695899 


47 


•466878 


•669203 47 


•481724 


•682799 


47 


•496729 


•696119 


48 


•467124 


•669432 43 


•481973 1 


-683G23 , 


48 


496980 


•696339 


49 


•467870 


•669661 49 


•482222 j 


•683247 


49 


•497282 


•696559 


50 


•467616 


•669889 ii 50 


•4S2471 1 


•683471 


50 


•497483 


•696778 


51 


467862 


•670118 ' 51 


•482720 ' 


•683695 


51 


•467734 


•696998 


52 


•46S1C9 


•670347 i 52 


482969 i 


•683919 


52 


497986 


-697217 


63 


•468855 


•670575 : 53 


•483218 i 


•684143 


53 


•498237 


•697436 


54 


•468601 


•670804 M 54 


•488467 i 


•684367 


54 


•498489 


•697666 


55 


•468848 


•671032 :; 55 


•483716 i 


•684590 


55 


•498741 


•697875 


56 


•4C9 94 


•671260 jl 56 


•483965 1 


•684814 


56 


498993 


-698094 


57 


•469841 


•671488 !: 57 


•484214 1 


-685037 


67 


•499244 


•698313 


58 


•469: S7 


•671716 j! 58 


•484463 


•685261 


58 


•499496 ; 


•698^2 


59 


•469884 


•671945 


59 


-484713 i 


•685484 


59 1 


499748 i 


•698751 


60 


•470C81 


•6721T3 


60 


•484962 j 


•6S5708 1 


60 1 


•500000 j 


•69S970 









ExTEr.NAL Secants. 






455 


57 DEGREES. 


' Min. 


58 DEGREES. 

1 Nat. No. ! L..gari;l.m. 


59 DEGREES. 


Mri. 


1 N^.t N,) 


1 L...a,.i,l,.n. 





Nat No 


Logarithm. 





0-833)78 


! 9^922247 


■ 


■887083 


9 947963 


©•941604 


9^973868 


1 


•3369 M 


i •922674 


1 


•887959 


•948393 


1 


•942544 


•974332 


2 


•S37725 


•923101 


2 


•888339 


•948833 


2 


•943486 


•974735 


3 


•83355) 


•923529 


3 


•889723 


•949253 


3 


•944429 


•975169 


4 


•839375 


923956 


4 


•89 )601 


•949683 


4 


•945373 


•975603 


5 


•84)2)2 


•934384 


5 


•891484 


•950114 


5 


•946317 


•976037 


6 


•8U029 


•924811 


6 


•893368 


•950544 


6 


•947263 


•976471 


7 


•841357 


•924238 


7 


•893353 


•950975 


7 


•948210 


•976935 


8 


•812636 


•925666 


8 


•894139 


•951405 


8 


•949158 


■977339 


9 


•8f3516 


•923093 


9 


•895)26 


•951836 


9 


•950107 


•977773 


10 


•844348 


•92 5521 


10 


•895914 


•952266 


10 


•951058 


•978207 


11 


•84513) 


•923949 , 


11 


•898802 


•952697 


11 


•9520r9 


•978641 


12 


•843)12 


•937377 


12 


•897693 


•953127 


12 


•952931 


•979075 


13 


•8468 L6 


•9278 M ^ 


13 


•893583 


•953558 


13 


•953915 


•979510 


14 


•817631 


•928232 i 


14 


•899475 


•953939 


14 


•954870 


•979944 


15 


•848516 


•923360 I 


15 


•90:)363 


•954420 


15 


•955S26 


•983379 


13 


•819353 


•929)88 


13 


•931263 


•954851 


U 


•956782 


•983813 


17 


•85)19) 


•939516 i 


17 


•932158 


•955283 


r 17 


•957743 


•931248 


IS 


•351023 


•929944 \ 


IS 


•933-53 


•955713 


18 


•958699 


•931682 


19 


•851837 


•933372 ; 


19 


•933949 


•956144 


19 


•959659 


•932117 


23 


•852707 


•93)800 I 


23 


•934347 


■956575 


20 


•960621 


•932552 


21 


•853843 


•931223 i 


21 


•9)5746 


•957006 


21 


•961583 


•9S29ST 


22 


•854393 


•931653 


23 


•936645 


•957437 


22 


•962546 


•9S3422 


23 


•855233 


•932)85 ; 


23 


•937546 


•957869 


23 


•963511 


•983857 


24 


•856)77 


•932513 i 


24 


•918448 


•958300 


: 24 


•964477 


•934292 


25 


•853921 


•932341 : 


25 


•9J9351 


•958732 


■ 25 


•965444 


•984727 


26 


•857767 


•933369 


26 


•91C255 


•959163 


' 26 


•966411 


•985162 


27 


•853314 


•933793 i 


27 


•911160 


•959595 


! 27 


•937380 


•985597 


2S 


•859131 


■934326 i 


23 


•912366 


•960026 


' 28 


•968350 


•986033 


29 


•83)31) 


•934655 ' 


29 


•912973 


•960458 


■ 29 


•969322 


•956468 


3i) 


•831159 


•935)83 : 


33 


•913381 


•960890 


' SO 


•970294 


•986903 


31 


•832 );}9 


•935512 1 


31 


•914790 


•961321 


i 31 


•971268 


•987339 


32 


•832S3) 


•935941 


32 


•915700 


•961753 


32 


•972242 


•9S7775 


33 


•833713 


•936339 


33 


•916611 


•962185 


33 


•973218 


•988210 


34 


•834535 


•933793 


34 


•917523 


•962617 


' 34 


•974195 


•988646 


35 


•83512) 


•937337 


35 


•918436 


•963049 


i 35 


•975173 


•089082 


36 


•833275 


■937656 i 


36 


•919353 


•963481 


1 36 


•976153 


•989518 


37 


•867131 


•938)85 > 


37 


•930365 


•963913 


' 37 


•977133 


•9-9954 


33 


•867937 


•938514 1 


88 


•931183 


•934345 


' 38 


•971815 


•990390 


39 


•833315 


•938343 1 


39 


•922 j99 


•984777 


; 39 


•979097 


•99 '826 


40 


•839704 


•339371 1 


43 


•923;)17 


•935210 


40 


•980081 


•991262 


41 


•871)534 


•9393)1 


41 


•923937 


•965642 


41 


•981066 


•091698 


42 


•871435 


•94323) 


42 


•924357 


•966075 


42 


•982052 


•91)2134 


43 


•87323) 


•94)659 


43 


•925778 


•966507 


43 


•983039 


•99-2571 


44 


•873148 


•911083 1 


44 


•923701 


•960940 


44 


•984;;27 


•993007 


45 


•874)12 


•911517 


45 


•937624 


•967372 


45 


•985017 


•99344.4 


46 


•874377 


•941947 


43 


•938549 


•967805 


46 


•986008 


•9035 8) 


47 


•875742 


•942376 


47 


•929475 


•938238 


47 


•987000 


•994:317 


48 


•8763)3 


•9433)6 


43 


•930401 


•968670 


48 


•987993 


•994754 


49 


•877475 


•943235 


49 


•931329 


•969103 


49 


•988987 


•995191 


50 


•878344 


•913365 


50 


•933258 


•969536 


50 


•9S9932 


•095027 


51 


•879213 


•914394 


51 


•933188 


•969969 


! 51 


•99.979 


■996064 


52 


•83)'.)8'> 


•91 152 4 


52 


•934119 


•970402 


52 


•991977 


•996501 


53 


•88)954 


•914953 


53 


•935050 


•970835 


53 


•992975 


•996938 


54 


•831827 


•945333 


54 


•935983 


•971268 


54 


•993975 


•097376 


55 


•8827i)) 


•945813 


55 


•936917 


•971701 


55 


•994976 


•997818 


56 


•833574 


•943243 


56 


•937853 


•972135 


56 


•995978 


•998250 


57 


•881419 


•946673 


57 


•938789 


•972568 


^ 57 


•996983 


•998687 


5S 


•835325 


•947103 


58 


•939726 


•973001 


58 


•997987 


•999125 


59 


•8362 )2 


•947533 


59 


•940664 


•973435 


59 


•998993 


■999562 


60 


•887080 


•947963 


60 


•941604 


•973868 


. 60 


10 00000 


1 •0000000 



NATURAL SINES AND TANi^ENTS, 



TO EVERY DEGREE AND MINUTE OF THE QUADRANT. 



EXTENDED TO SEVEN PEACES OF DECIMALS, 



ag 



453 



Natural Sine^-, 



/ 


(.• j 1° 2' 


3° 4^ 6" 


0° 1" 







000 00001017 4524 C34 8995 052 3360'0e9 7565'C87 1557104 5285121 8G93 60 


1 


20,9 74S2 035 1902| 6264:070 0467 44551 8178122 1581; 59 


2 


5S1S018 03411 48r;9i 9109' 3368' 7353105 1070i 4468 58 


3 


8727 


32491 7716 053 2074 6270;0S8 G25li 39631 7355 57 


4 


001 1636 


6158 C36 0623! 4979 9171 1 3148! 6856 123 0241156 


5 


4544 


9366 


35301 78S3071 2073 6046 9748; 3128 55 j 


6 


7453 019 1974 


6437 054 C788I 4974! S9431G0 264l! 6^:15; 54 j 


7 


002 0362 


4S83 


9344 


3693| 7876 089 18401 5533! 89 1| 53 1 


8 


3271 


7791 '037 2251 


6597,072 0777! 4738| 8425 124 1788! 52 i 


9 


6180 


020 0699 


5158 


9502 


3678j 7635,107 1318 4674 


51 


10 


9:;69 


36C8 


8065 


055 2406 


6580 C90 05321 4210 7560 


50 


11 


003 1998 


6516 


038 0971 


5311 


9481 


3429 7102 125 0446 


49 


12 


4907 


9424 


3878 


8215 


073 2382 


6326 9994 8332 


iS 


1.^ 


7815 


021 2332 


67S5 


056 1119 


5233 


9228108 28.-5 6218 


47 


14 


004 0724 


5241 


9692 


4C24 


8184 C91 2119, 5777! 9104 


46 


15 


3633 


8149 039 2598 


6928!074 10851 5016| 8669 126 1990 


45 


16 


6542 


022 1057 5505 


9832 


3986 7913:109 15j0j 4875 


44 


17 


9451 


3965 8411 


057 2736 


6887 ;092 0809 44521 7761 


43 


18 


005 236ii 


6873 040 0318 


5640 


9787! 3706 7343127 0646 


42 


19 


5268 


9781 4224 


8544075 2688 6602110 C234! 3531 


41 


20 


8177 


023 2690 7131 


058 1418 5589 9499 


3126 mQ 


40 


21 


006 1086 


5598 041 0037 


4352 8489 093 2395 


6017! 9302 


39 


22 


3995 


8506 


2944 


7256 076 1390 5291 


89.:;8 128 2186 


38 


23 


6904 


024 1414 


5850 


059 01601 4290 81871111799: 5071 


37 


24 


9813 


4322 


8757 


3064! 7190 094 1C83| 4089; 7956 


36 


25 


007 2721 


7230 042 1663 


5967,077 0C91 


3979! 7580 129 0841 


35 


26 


5630 


025 01381 4569 


8S71 


2991 


6875112 0471! 8725 


34 


27 


8539 


8046 7475 


060 1775 


5891 


9771 


3361 06(9 


33 


28 


008 1448 


5954 043 03S2 


4678 


8791 095 2666 


6252; 9494 


32 


29 


4357 


8862 


3288 


7582078 1691 


5562 


9142 130 2378 


31 


30 


7265 


026 1769 


6194 


061 04S5| 4591 


8458 113 2082; 52G2 


30 


31 


009 0174 


4677 


910J 


33S9j 7491 


096 1353! 49221 8146 


29 


32 


3083 


7585 044 2006 


6292 079 0891 


4248! 7812131 1(30 


28 


33 


5992 


027 0493 4912 


9196 3290 


7144114 C7C2, 3913 


27 


34 


8900 


3401 7818 


062 2C99 0190 


097 0(89, 3592; 6797 


26 


35 


010 1809 


6309 045 0724 


50C2 9090 


2934 6482; 96S1 


25 


36 


4718 


9216 


3630 


7905080 19S9 


5829' 9372 132 2564 


24 


37 


7627 


028 2124 


6536 


063 0808! 48S9 


8724115 2261! 5447 


23 


38 


Oil 0535 


5032 


9442 


• 3711 77S8C98 1619' 5151| 8330 


22 


39 


3444 


7940 


046 2347 


6614 0810687! 4514| 8040133 1213 


21 


40 


esi-fj 


029 0847 


5253 


9517 35871 7408116(929! 4(96 20 


41 


9261 


3755 


8159 


064 2420 6486 0S9 03081 8818; 6979119 


42 


012 2170 


6602 047 1065 


5323 9385 


3197 6707! 98G2. 18 


43 


5079 


9570 3970 


8226 082 2284 


6(921 9596134 2744: 17 


44 


7987 


030 2478 6876 


065 1129! 5183 


8986117 2485! 5627' 16 


45 


013 0898 


5385 9781 


40311 8C82:i00 1881! 5374j &5r9 15 


46 


3805 


8293 043 2687 


G934 083 0981 4775^ 8263135 1£92: 14 


47 


6713 


031 12)0 5592 


9836, 3880 7669 118 1151 4274 13 


48 


9622 


41G8 8498 


066 2739 6778101 0563! 4040 7156: 12 


49 


014 2530 


7015 049 1403 


5641 96771 3457! 6928136 0038 11 


50 


5439 


9922 


4308 


8544 084 2576 635l| 9816J 2919, 10 


51 


8348;032 2830 


7214 


067 1446: 5474! 9245119 27C4 58' l| 9 


52 


015 12561 5737 


050 0119 


4349 8373,1(2 2188: 5593! 6083: 8 


53 


4165! 8344 


3024 


7251 085 1271 


5C82i 8481 137 1564; 7 


54 


7073 033 1552 


5929 


068 0153, 4169 


7925 120 1368' 4445; 6 ' 


55 


9982 4459 


8835 


3055 7067 


10a0819i 4256! 7827| 5 


56 


016 2890 7366 0511740 


59571 9966 


8712! 7144138 0208i 4 


57 


5799 034 0274 4645 


8859 086 2864 


6605121 0031| 30891 3 


58 


8707 3181 755( 


069 17611 5762 


9499; 2919 5970 2 


59 


017 1616 6088 052 0455 


4663! 8660!l04 2o92 


58061 8850 1 


60 


4524 8995 3360 


7565 087 1557 5285 


8693139 1731 


' 


89= 1 88° 1 87° 


80° 1 85° 84° 


83° 1 r2° 1 ' 


Natural Cosines. 



K'atural Tangents. 



459 



' 


0° r 


2° 





4° 


5° 6° r ' 





000 000( 


jl7 4551 


034 9208:052 407^^ 0C9 9263 087 4SS7 1C5 1042 122 7846! 60 


1 


2909 


746C- 


035 2120; 6995 070 2191; 7S13J 8983123 0798 59 


2 


5813 


D18 0870 


50381 9912; 5115:088 0749; 6925 3752: 58 


8 


8727 


323;) 


7945.053 2829, 8038! 3631: 9366! 6705| 57 


4 


001 1636 


6190 


036 0858 


5746;071 C96li 6612106 2808! 9658! 56 


5 


454i 


9100 


8771 


8663 


£8351 9544; 5750,124 2612! ^^ 


6 


7453 


019 2010 


6683 


054 1531 


68C9 089 2476! 8692] 55661 54 


T 


0G2 0SG2 


4920 


9596 


4498 


9733 


54(8107 1634| 8520 53 


8 


3271 


7830 


037 2509 


7416 


072 2657 


8341 4576,125 1474! 52 


9 


618'; 


020 0740 


5422;055 0333 


5581 


090 1273 7519 


4429, 51 


10 


9:;89 


3650 


8335 


3251 


85 5 


4206 1C8 0462 


7334 


50 


11 


003 19;}3 


6560 


038 1243 


6169 


073 1430 


7138! 3405 


126 0339 


49 


12 


4907 


9470 


4161 


9087 


4354 


C91 0O71 6348 


8294 


48 


13 


7816 


021 2383 


7074 


056 2005 


7279 


3004 9291 


6249 


47 


14 


004 0725 


5291 


99S3 


4923 


074 0203 


59:8 iC9 2234 


9205 


46 


15 


3634 


82.1 


039 2901 


7341 


3128 


8371 j 5178 


127 2161 


45 


16 


6542 


022 1111 


5814 


057 0759 


6053 


092 1804 8122 


5117 


44 


IT 


9451 


4021 


8728 


3678 


8979 


4788 110 1066 


8073 


43 


18 


005 2360 


6932 


040 1641 


6596 


075 1904 


7672 4010 


128 1030 


42 


19 


52C9 


9842 


4555 


9515 


4829 


093 0606 6955 


8983; 41 


20 


8178 


023 2753 


7469 


G58 2434 


7755 


^40 9899 


69431 40 


21 


006 1187 


5663 


041 0383 


5352 


076 0680, 


6474111 2344 


990o| 39 


22 


3996 


8574 


3296 


8271 


3606 


94C9 


5789 


129 2353! '"^8 


20 


6905 


024 1484 


6210 


059 1190 


6532 


094 2344 


8734 


5815* 37 


24 


9814 


4395 


9124 


41(9 


9458 


5273 


112 1680 


8773 


86 


25 


007 2723 


7305 


042 2038 


7029 


077 2384 


8213 4625 


180 1781 


85 


26 


5632 


025 0216 


4952 


9948 


5311 


095 1148! T571 


4690 


34 , 


27 


8541 


3127 


7856 


060 2307 


8237 


4084,113 0517 


7648 


33 


28 


008 1150 


• 6U38 


043 0731 


5737 


078 1164 


7019 3463 


181 0607 


82 


29 


4360 


8948 


3095 


87v;6 


4090 


9955 


6410 


8566 


31 


80 


7269 


026 1S59 


6609 


061 1626 


7017 


:.96 259( 


9356 


6525 


80 


81 


OCO 0178 


4770 


9524 


4546 


9944 


5826 


114 2303 


9484 


29 


82 


3087 


7681 


044 2438 


7466 


079 2871 


8763 


5250 


182 2444 


28 


88 


5996 


027 0592 


5353 


062 0SS6 


5708 


097 1699 


8197 


5404 


27 


34 


8905 


8503 


8265»' 8306 


8726 


4635 


115 1144 


8364 


26 


85 


010 1314 


6414 


045 1183 


6226 


080 1653 


7572 


4092 


183 1324 


25 


36 


4724 


9325 


4097 


9147 


4581 


098 G509 


7039 


4285 


24 


87 


7633 


028 2236 


7012 


063 2067 


75C9 


3446 


9937 


7246 


23 


88 


Oil 0542 


5143 


9927 


4933 


081 0437 


6883 


116 2936 


134 0207! 22 


39 


3451 


8':59 


046 2812 


79.8 


8365 


9320 


5334 


31G8! 21 


40 


0361 


029 C970 


5757 


064 CS29 


0293 


099 2257 


8332 


6129 20 


41 


9270 


88S2 


8673 


3751 


9221 


5194 


lit 1781 


9.91 


19 


42 


012 2179 


6793 


04T 1588 


6671 


082 2150 


8133 


4730 


135 2053 


18 


48 


5088 


9705 


4503 


9592 


5078 


100 1071 


7679 


5015 


IT 


44 


7998 030 2616 


7419 


065 2513 


8007 


40C9|118 (.623 


7978 


10 


45 


013 0907 5528 048 0834 


5435 


083 Ca;36 


694-7 


3578 


136 C940 


15 


46 


3317 8439 


3250 


8356 


8365 


9336 


6528 


3903 


14 


47 


6726 081 1351 


6106 


066 1278 


6794 


101 2824 


9478 


0806 


13 


48 


9635{ 4263 


9082 


4199 


9723 


57()3 


119 2428 


9330 


12 


49 


014 2545 7174 


049 1997 


7121 


084 2653 


87C2 


5373 


137 2793! 11 


50 


5454 032 CCSG 


4913 


067 0043 


5533 


102 1641 


8329 


5757; 10 


51 


8364 2998 


7829 


29G5 


8512 


4580 


120 1279 


8721 


52 


015 1273 591(1 


050 0746 


5887 


085 1442 


7520 


4230 


138 16S5 8 


58 


4183 8822 


3662 


83,9 


4372 


103 04601 7132 


4650 7 


54 


7C93 033 1734 


6578 


068 1732 


7302 


3399 121 0108 


7615 6 


55 


016 0002 4646 


9-195 


4654 


083 0233 


6340 3085!l89 C5S0| 5 


56 


2912 7558 0512411 


7577 


3163 


9280 6036 


3>545 4 


57 


5821034C471 5328 


069 0499 


6094 


104 2220 8988 


6510 8 


58 


87311 3383 8244 


3422 


9025 


5161 122 1941 


9476 2 


59 OIT 1641 G20:>m 1161 


6345 


087 1956 


8101 4893140 2442! j 


60 4551 9:>;.8 4078 


9263 


4S87 


1C5 1042! 7846 54: S' 


89° 8b'' 1 87^' 


8tr 


Po^ 


84^ 1 ^r r^^i' 1 ' 


NATUEAI4 Cotangents. 



460 Natural SfNKS. 










' 


8° i 9° 1 10° 1 U° 1 \r \ 1S° ! ]4° i 15' 







189 1731156 4:345173 6482190 Sr,90 207 9117 224 9511241 9219 258 8190 


60 


1 


4612 7218 9346 191 ( 945 2C8 1962 225 2345 242 2* 41 259 ICOO 


59 


2 


7492157 0091174 2211 8801; 4807! 5179 4863 8810 


58 


3 


140 0372 2963 5075' 6656' 7652' 8013 76S5 6619 


57 


4 


8252 5836 79S9 9510 2C9 0497 226 (846 243 C507 942S 


56 


6 


6132: 8708 175 G803 192 2365 3841 8680, 8829 260 2287 


55 


■ 6 


9012158 15811 3667 522o! 61S6 6518, 6150 5045 

141 1892: 445a 6581 8074; 9.:80 9846' S97i' 7858 

4772 7825! 9395193 0928 210 1874 227 2179 244 1792 261 0662 


54 


7 


53 


8 


52 


9 


7651159 0197176 2258 87821 4718f 5C12. 4613, 8469 


51 


10 


142 0531 3069 5121 6636i 7561 7844' 7483 6277 


50 


11 


3410, 5940l 7984 9490 211 04C5 22S G677 245 (254' 9rS5 


49 


12 


62S9 8812177 (847194 2344 


8248 85(9 3074 262 1892 


48 


13 


9168 160 1683' 8710 5197 


6(91 68411 5894 4699 


47 


14 


143 2047' 4555 6573' 8050 


8934 9172 8713' 7506 


46 


15 


4926! 7426 9435195 0908 212 1777 229 2CC4 246 1538 263 0812 


45 


16 


7805;i61 0297178 2298 8756 4619 


48851 4352, 8118 


44 


17 


144 0684; 3167. 5160 66091 7462 


7666! 7171| .^925 


43 


18 


8562; 6088; 8022' 9461210 C3C4 


230 0497! 9990; S73( 


42 


19 


&440i 89G9179 (884196 2814 8146 


8828 247 28(9 264 1586 


41 


; 20 


9819:162 1779, 3746 5166 59SS 


6U9, 5C27, 4842 


40 


21 


145 21971 4650 8607 8018 SS29 


89S9 8445; 7147 


89 


; 22 


5075i 7520 9469197 (870 214 1671 


231 1819 248 1263 9952 


88 


1 23 


7953163 0890 ISO 2880: 3722 4512 


4649, 4(81265 2757 


37 


! 24 


146 0880! 8260; 5191 ' 6573 7858 


7479 f899. 5561 


36 


25 


37C8' 6129i 8052 9425 215 0194 


232 C8C9' 9716 8866 


^ 


: 26 


6585: 6999 :131 (913198 2276 3085 


3186 249 2588 266 117( 


34 


27 


9463164 1868 ; 8774, 5127 5876 


^967i 5850, 3978 


33 


28 


147 2340{ 4738' 6635; 7978 8716 


8796! 8167] 6777 


32 


: 29 


5217; 7607! 9495199 0829 216 1556 


283 1625'250 (984! 9581 


31 


i 30 


8094 165 0476 182 2855 86791 4?96 


4454 


8800 207 2884 


ao 


1 31 


148 0971; 8845! 5215 6530' 7236 


7252 


6616, 5187 


29 


i 32 


3848 


6214 8075: 9880.217 0076:234 0110 


9482: 7989 


28 


; 33 


6724 


9082 183 C935 200 22301 2915 


2938 


251 2248 268 0792 


27 


1 S4 


9601 


166 1951: 3795 5080! 5754 


5766 


5(63 8c94 


26 


1 35 


149 2477 


4819; 6654! 7930; 8598 


5594 


7879; 6896 


25 


• 36 


5853 


7687; 9514;2C1 0779 218 1482 


235 1421 


252 0694! 9195 


24. 


37 


8280 


167 0556 184 2378 


3629! 4271 


4245 


85(8 269 2000 


231 


38 


150 1106 


8428: 5232 


. 6478; 7110 


7075 


68231 4801 


22 « 


39 


89811 6291'- 8C91 


9827 i 9948 


99(2 


91871 76C2 


21 


40 


6857 0159,185 C949'2C2 2176219 27S6;230 2729 


253 1952 270 C4C8 


20 


41 


9788168 2026! 8808 50241 56241 5555 
151 26C8 4894; 6666 7873' 8462 8881 


4766; 3204 


19 


1 42 


7579 60C4 


18 


43 


5484 7761'; 9524 208 0721 ;220 18C0|237 1207 


254 0£93! 8805 


17 


44 


8359169 0628186 2382 8569 4187 


4083 


8206 271 1605 


16 


45 


152 1284 8495; 5240 * 6418 C974 


6859 


6019 4404 


15 


46 


41(9; 6862': 8098 9265 9511 


9684 


8882 7204 


14 


47 


69S4i 9228187 C9562C4 2118 221 2648 23S 2510 


255 1645272 0008 


13 


48 


9S5S170 2095; 8813 49611 5485! 5885 


44581 2802 


12 


49 


153 27381 49611 6670 7808' 832ll S159 


7220 5601 


11 


i 50 


56^7| 7828^ 9528 205 •0655222 1158 239 (984 


256 0C82 8400 


10 


51 


84821710694188 28851 £502i 8994i 8SC8 


2594 273 1198 


9 


52 


154 1356 3560; 5241 6349! 6S80: 6688 


5705 8997 


8 


53 


4230 6425; 8098; 9195; 9666! 9457 


8517 6794 


T 


54 


7104 9291 1S9 (954206 2042 223 2501 240 2280 


257 1328 9592 


6 


55 


9978.172 2156: 881l| 48SS 5887. 5104 


4139 274 2390 


5 


56 


155 2S51| 5022; 66671 7784 8172! 7927 


6950; 5187 


4 


5T 


5725! 7887, 9523 207 0580,224 1007 241 U751 


9760i 6984 


3 


58 


8598173 0752190 2379, 3426 8842; 3574 


258 2570275 0781 


2 


59 


156 1472 8617 5284 6272; 6676 6396 


5351 ; 3577 


1 


60 


4845 6482 8090 9117' 9511! 9219 


5190^ 6374 





' 


Sr ; 80' 79' 1 78' ! 77° \ 76' 


75^ 1 74° 


/ 








ISTatukai 


C0SI>'ES 


5. 











Natural Tangents. 






4G1 


' 1 8° 1 9° 1 10° 1 11° 12° 13° 1 14° 1 15° ' 1 





140 5408158 3844176 3270194 88^!8 212 5566 280 8(582 249 3-280 267 9492 60 t 


1 


88751 6826 6269' 6822 8606 2311746 6870 268 2610 59 : 


2 


141 1342 98; 9 9269 9S41 213 1647, 4S11 94601 5728 58 


3 


4808159 2791177 2269195 28611 4688 7876 250 2551 8847 57 


4 


7276| 5774 5270 58811 7780 232 C941 5642 269 1967 56 


5 


142 02481 8757^ 8270 89.1214 0772, 4007 8784: 5l*87 55 


6 


3211160 174)178 1271196 1922 3814 7078 2511826 8207 54 


7 


6179 4724 4273! 4948 6857 233 01401 4919 270 1828 53 


8 


9147 77^.8 72741 7964 99001 3207| 8012. 4449 52 


9 


143 2115161 0692179 0276197 C936 215 2944' 6274 252 1106| 7571 51 


10 


5084! 3677 3279, 4008' 5988! 9:342! 4200 271 0694 50 


11 


80581 6662 628l| 70811 9v32 234 24101 7294: 3S17 49 


12 


144 1022' 9647 92S4 198 0058 216 2077| 5479 253 0889' 694':) 48 


13 


8991162 2682 18J 2287 


30761 5122^ 8548 3484 272 0064' 47 1 


14 


693l| 5618 5291 


6100i 8167 235 1617 6580J 8183 46 1 


15 


998l! 3603 8295 


9124 217 1218 4687 9676 6318 45 


16 


145 29)1163 159)181 1299199 2148 42c9' 7758 254 2773' 9488 44 


17 


5872 4576 48j8i 5172' 7806 236 0829: 5870 273 2504 43 


18 


8342 7563 78U8' 8197 218 0858 8900 8968| 569;) 42 


19 


146 1813164 0550182 0813 200 1222 8400 6971255 2066| 8817' 41 


20 


4784 3537, 38191 4248 6448 237 0044| 5165 274 1045 40 


21 


7756 6525 63241 7274 9496J 3116' 8264 5072 89 


22 


147 0727 9513 9380 23103)0 219 2544 6189 256 1863 820188 


23 


3699 165 25)1 183 2887 38271 55981 9262! 4463 275 1830' 37 i 


24 


6672| 5489 5348 68541 8648 238 2886, 7564 4459 ! 36 1 


25 


9(344' 8478 8850 9381223 1692: 5410 257 0664 7589 35 i 


26 


148 2617166 1457184 1358 202 24)91 4742 8485 


8766 276 0719 34 ' 


27 


5593 4456 43651 54871 7798 239 1560 


6868] 3850 33 ' 


28 


8568 7446 7873, 8465 221 C844! 4685 


9970 6981' 32 1 


29 


149 1536 167 0436 185 0382 203 1494 


3895! 7711258 3073 277 0113 31 ; 


30 


4510 


3426 3893 4528 


6947240 0788 6176 


3245 30 i 


31 


7484 


641 7i 6899 7552 


9999 3864 928 ) 


6878 29 ' 


32 


150 0458 


9407 9409 204 0582 222 305l| 6942 259 2884 


9512 28 : 


33 


3488 168 2393 186 2418 3612 6104 241 0019 5488 278 2646 27 


34 


64081 589J, 5428 6648 9157 3r,97 85981 5780 26 


35 


9383| 8831' 8489 9674 223 2211 6176 260 16991 8915 25 


36 


1512358169 1378187 1449 205 2705 5265 9255J 48 5 279 2050 24 


37 


58881 4366 4460 5787 8819 242 2834 791l| 5186 23 ' 


38 


8809| 7858 7471 8769 224 1874 5414 261 1018! 8322 22 \ 


39 


152 1285170 0351188 0483 206 18311 44291 8494i 4126 280 1459 21 1 


40 


4262 


3844 3495 4884 7485 243 1575| 7284 4597, 20 | 


41 


7288 


6333 6507 7867 225 05411 4656 262 0342 7735 19 i 


42 


153 0215 


9381 9520 237 09)0 


3597! 7737 3451 281 C873 18 j 


43 


3192 171 2825 139 2588| 3934 


6654 244 0819, 6560 4012, 17 j 


44 


6170 5823, 55461 6908 


97111 89021 9670 7152 16 1 


45 


9147 8814 8559 208 0008 226 27691 6984 263 2780282 0292 15 i 


46 


154 2125172 1309193 1578 3j83j 5827,245 0068! 5891 


3482 14 


47 


5103 4804 4587 6078' 8885 


31511 9302 


65781 13 1 


48 


8082 7330' 76)2 91(.9 227 1944 


6286 264 2114 


97151 12 I 


49 


155 1061 173 0296 191 0617 209 2145! 5003 


98201 5226 283 28571 11 1 


50 


4040 3292 * 3682 518 li 8063246 2405| 8889 5999! 10 ! 


51 


7019 6283' 6648 8218 228 1123 5491265 1452 9145i 9 ! 


52 


9998 9285 9664 210 1255 4184 85771 4566 284 2286 


8 ! 


53 


156 2978 174 2282 192 268) 4293 7244 247 1668' 7680 5430 


T t 


54 


5958 5279 5696 7331 1229 0306! 4750 266 0794 8575 


C 1 


55 


89891 8277; 8718 211 0369 


83671 7887 8909 285 1720 


5 ! 


56 


157 1919175 1275198 1781! 3407 


6429 248 0925 7025 


4866 


4 


57 


4930 4278 4743 6446 


6492 4018 267 0141 


8012 


3 


58 


7831 7272 7766| 9486:230 25551 71 (!2 8257 


286 1159 


2 


69 


158 0863 176 0271 194 0784 212 2525 


5618 249 0191 6874 


4306 


1 


60 


8844| 3270| 8808 5566 


8682i 8280 9492 


7454 





( 


SV 1 80'^ 1 79" 78° 


77^ 


7(- 1 75° 


74^^ 


1 




-Nr*.«T«AT f 


/OTANGE> 













T8. 







462 


Natural Sines. 








' 1 16° 17° 1 18° 19° 20° : 21° \ 22° I 23° i ' 





275 6374 292 8717 3C9 0170 325 5682 342 0201358 3679 374 6066 390 7311! 60 


1 


9170; 6499 2936 84:32 2935 6395 8763 0959^ £9 


2 


276 1965 9280 5702 326 11 S2 5668 9110 375 1459 391 2666' 58 


3 


4761 293 2061 846S 3932 8400 359 1S25 4156 5343 57 


4 


7556: 4842 310 1284 6681 343 1183 454') 6852 8019 56 


5 


277 0352 7623 3999 9430 3S65' 72.54 9547 392 0695 55 1 


6 


3147 294 0403 6764 327 2179 6597 9968 376 224:3 3371 


54 1 


T 


5941' :31S8 9529 4928 9:329 360 26S2 4938 6047 


53 1 


8 


8786^ 5963 311 2294 7676 844 20601 5895 76321 8722 


52 


9 


278 1530; 8743 505S:323 0424 4791! 8108:577 0327 393 1397 


51 


10 


4:324 295 1522 7S22 3172 7521 361 0S21 3021 


4071 


50 


11 


7118' 4302 312 05S6: 5919 345 0252; 8534 5714 


6745 


49 


12 


9911! 70S1' :3:349 8666 2982: 6246 84C8 


9419 


48 


13 


279 27C4I 9859 6112:329 1413 5712^ 8958 378 1101 394 2G98 


47 


14 


M97 296 268S 8S75 4160' 8441 362 1669 3794^ 4766 


46 


15 


8290; 5416 313 16:38 6906:346 1171 4380i 6486i 7439 


45 


16 


280 1088! 8194 4400i 9658 3900 7091' 9178 395 0111 


44 


IT 


8875 297 0971 7163 3:30 2:398 6628' 9802 379 1870 2783 


43 


18 


6667; 3749 9925 5144 9357 363 2512, 4562 5455 


42 


19 


9459; 6526 314 26S6' 7889 347 20851 5222' 7253^ 8127 


41 


20 


2812251 9:308 5448 8:310634 4812' 79:32 9944 396 0798 


40 


21 


5042298 2079 8209 3379' 7540 364 0641380 2634 3468 


39 


22 


78:33: 4856 315 0069 6123 348 0267 8:351; 5324 6139 


38 


23 


282 0624 7632 3730' 8867; 2994 6059 8014 8509 


37 


24 


3415 299 0408 649ij 332 1611 ! 57201 8768:3810704:397 1479 


36 


25 


6205, :3184 9250; 4355 8447 865 1476, 3:393| 4148 


35 


26 


8995; 5959:316 2010 7098 349 1178 4184 6082! 6818 


34 


27 


283 1785' 8784 4770! 9S41j 3898 6891 8770i 9486 


33 


28 


4575 300 1509 7529 3:33 2584! 6624 9599 382 1459 398 2155 


82 


29 


7364; 4284 317 0288; 5:326 9349:366 2:306, 41471 4823 
284 0153! 7058 3047'! 8069 :350 2>;74l 5012 6834! 7491 


31 


30 


30 


31 


2942^ 9S32 5805 334 0810| 4798 7719 9522 399 0158 


29 


32 


57313912606 856:31 3552 7523 367 0425 383 2209 2825 


28 


38 


8520 5380 818 1:321- 6293:3510246! :3130! 4895 5492 


27 


34 


285 1308: 8153 4079 9 ^34' 2970^ 5836 7582 8158 


26 


35 


4096 302 0926 68:36 335 1775; 5698! 8541384 0268 400 0825 


25 


36 


6SS4 3699 9593! 4516 8416363 1246 29531 3490 


24 


37 


9671 6471319 2:350' 7256 352 11:39 3950 5689! 6156 


23 


38 


286 2458: 9244 5106' 9996' 3S62! 6654' 8324! 8821 


22 


39 


6246 303 2016 7863 336 27:35| 6584 935S 385 1008 401 1486 


21 


40 


8032: 4788 320 0619 5475; 9306:369 21)61^ 3693! 4150 


20 


41 


287 0819; 7559 3374 8214353 2027! 4765^ 6377; 6814 


19 


42 


8605:304 0331 6130:8:37 0958! 4748! 746S! 9060| 9478 


18 


48 


6891! 3102, 8885 369li 7469 370 0170 386 1744 402 2141 


17 


44 


9177! 5872 321 1640' 6429 354 0190 2872; 4427} 4804 


16 


45 


288 1963! 8643 4:395! 9167 2910 5574i 7110 7467 


15 


46 


474S305 1418 7149 838 19 5 56:30i 8276 9792403 0129 


14 


47 


7538 4188 9903 4f>42- 8:350 371 0977 387 2474; 2791 


13 


48 


289 0318! 6958 322 2657' 7:379:355 1070! 3678i 5156 5453 


12 


49 


3103' 9728 5411^839 0116! 3789' 0379^ 7887 8114 


11 


50 


5SS7 306 2492 8164 2S52i 650S! 9079 388 0518 404 0775 


10 


51 


8671| 5261 823 0917: 55S9i 9226^372 17S0| 8199 3436 


9 


52 


290 1455 8030 3670; 8325 :356 1944, 4479! 5880 6096 


8 


53 


4239:307 0798; 6422:^0 1060 4662! 7179' 8560 8756 


7 


54 


7022J 3566 9174 3796 7380^ 9S78 389 1240 405 1416 


6 1 


55 


9805 6334 324 1926 6531:357 0097:373 2577! 8919 4075 


5 


56 


291 2583' 9102| 4678 9265! 2814 5275' 6598 6734 


4 ! 


57 


5371308 1869! 7429:341 2000 j 5581 7978: 9277 9393 


3 1 


58 


8153 46:36 325 0130 47:34' 8248 374 0671 890 1955 406 2051 


2 1 


59 


292 09:35 7403: 2931 7468:358 0964^ 3369' 4683! 4709 


1 ' 


60 

/ 


3717:309 01 70' 5682:342 02Gl! 3679 6066 7311, 7366 
73° 1 72° 1 71' 1 70° 1 69° ! 68" : 67° ! 66° 







Natueal Cosines. 


j 









Natural Tangents. 






463 


' 


16° 


\r 


18° 


19° 20° 21° 


22° 


23° 


/ 



40 
41 
42 
43 
44 
45 
46 
4T 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



286 T454'305 73071324 9197,344 3276 363 9702 383 8640 404 0262 424 4748 

287 0602306 0488 325 2413 6530 364 2997 384 19781 3646' 8182 
37511 3670 56301 97851 62921 5317| 7031425 1616 
69001 6852 8848 345 3040| 9588 8656 405 04171 5051 

288 0050 307 0034'326 20661 6296 365 2885 385 19961 8804' 8487 
82011 32181 5284' 9553 6182 5337| 7191 426 1924 
6352 6402 8504 346 2810! 948(»| 8679 406 0579! 5361 
9503' 9586327 1724 6068 366 2779 886 2021 1 8968| 8800 

289 2655 308 27711 4944' 93271 6079 5364 7358 427 2289 
58081 59571 8165 347 2586| 9379, 8708 407 07481 5680 
8961 9143 328 13871 5846 367 2680 387 20531 4189 9121 

290 2114309 233o| 4610' 91071 59811 5898 75311428 2563 
5269! 5517 7833348 23681 9284' 8744'408 0924 6005 
84231 8705 329 1056; 5630 368 2587 388 2C9l| 4318 9449 

2911578 310 1893 428l| 8893 58901 5439 7713 429 2894 
47341 5083 7505 349 2156| 9195J 8787 409 1108 6389 
78931 8272 330 0731] 5420 369 2500 889 2136 4504 9785 

292 1047 311 14621 3957! 86851 58061 5486 7901 '430 8232 
42051 4653 7184 350 1950| 91121 8837 410 1299 6680 
7363 7845 331 04111 5216 370 2420 390 2189 4697 431 0129 

293 0521 312 10361 36391 8483 57281 5541 8097 3579 
36801 42291 6368 351 175o! 9036| 8894 411 1497 7030 
6839| 7422 332 00971 5018 371 2346 891 22471 4898 432 0481 
9999313 0616 3327i 82871 66561 5602 8300 

294 3160| 3810 6557 352 1556' 89671 S957 412 1703 
6321! 7005 9788; 4826 372 2278 392 2313 5106 
9483 314 0200 333 302o| 80961 55901 6670 8510 

295 26451 33961 6252 353 1368 89C3I 9027 413 1915 
5808 6593 94851 4640 373 2217 393 23861 5321 
897ll 9790 334 2719 79121 55321 5745 8728 

296 2135 315 29881 5953 854 1186| 88471 9105414 2136 
52991 6186 91881 4460 374 2163 894 24651 5544 
84641 9385'335 2424| 77341 5479| 5827| 8953 

297 1680 316 25851 5660 355 1010| 87971 9189 415 2363 
47961 5785 8896] 4286 375 2115 895 2552; 5774 
7962 8986336 2134' 75621 5483J 5916| 9186 

298 1129 317 21871 5372 356 C840| 8753i 9280 416 2598 
42971 5389 8610! 4118 376 2073 896 2645| 6012 
7465! 8591337 1850i 73971 5394j 6OIII 9426 

299 0634318 17941 5090 357 0676! 8716 9378 417 2841 
3803i 4998] 88301 3956 377 2038897 2746! 6257 488 2756 
6973! 8202 338 15711 7287| 5361 6II4! 9673 6224 

300 0144319 14071 4813 358 C51S' 8085 9483 418 8091 9693 
3315| 4613 8056! 3801 378 2010 898 2853i 65(9 489 3163 
64861 7819 339 1299i 70831 5885J 6224^ 9928 6684 
9658 320 10251 4543 359 0367| 866l| 9595419 8848 440 0105 



483 0840 
4295 
7751 

434 1208 
4665 
8124 

435 1583 
5048 
8504 

436 1966 
5429 
S893 

437 235' 
5823 
9289 



301 28311 4232 7787 
60041 7440340 1032 
9178 321 06491 4278 

302 2352| 3858 1 7524 
55271 7067 341 0771 
8703322 02781 4019 

303 1879, 8489; 7267 
5055 6700 342 0516 
8232 9912; 3765 

304 1410 323 3125| 7015 
4588j 6838 343 0266 
7767! 95521 3518 

305 0946324 2766! 6770 
4126; 5981 344 0023 
7307 91971 8276 

78° I 72° 71 



3651379 1988 399 29681 6769 8578 

69361 63151 6341 420 0190 7051 

360 0222! 8644 9715; 3613 441 0526 

8508 330 1973 400 8089 7036 4001 

67951 53021 6465 421 0460 7477 

36100821 8683 9841; 8885 442 0954 

8371381 1964 401 8218 731ll 4482 



6660| 5296 
99491 8629 

362 3240 882 1962 
65311 5296 
9823! 8631 

363 3115 883 1967 



6408 

9702 

70 



5303 

8640 

69 



6596 422 07881 7910 
9974| 4165443 1890 



402 8354 7594 
6734 423 1023 

408 0115 4458 
3496 7884 
6879 424 1316 

404 (.262, 4748 
68° I 67° 



4871 
8351 

444 1884 
5818 
88(2 

445 2287 
66^ 



Natural Cotangents. 



/r,4 



Natural Sines. 





24° 25° 26° 1 27 ° 1 28° 1 29° 30° 31° 


' 1 


1 


4J6 7366 422 6183|438 8711453 99j5 469 4716 484 8u96 500 0000 515 03811 60 


1 4J7 00241 S819i 6326 454 2497| 7-284 485 0640i 2519, 2874' 59 ! 




2 2S31 423 1455 


8940, 5088: 9852, 8184 5037 586T. 58 ! 




8 5337 409J 


489 1553 7679 470 2419 57271 7556! 7859 57 ! 




4 7993 6725 


4166 455 02691 4986| 8270 501 0078 516 08511 56 i 




5 408 06 i9 9360 


6779 28591 7553 486 0812 25911 2842' 55 | 




6 


3305 424 1994 


9392 5449 471 0119i 83541 5107| 5^388' 54 ! 




7 


5960 


4628 


440 2004' 8083 


2685 5895' 7624! 78241 53 




8 


8'il5 


7262 


4615 456 0627 


5250 8436 502 0140 517 OSU' 52 ! 


I 


9 


4j9 1269 


9395 


7227, 8216 


7815 487 09771 26551 2804' 51 1 


1 


10 1 8923 


425 2523 


9833 5804 472 08S0 


3517 5170 5293' 50 


1 


11 


6577 


5161 


441 2448 8392 


2944 


60571 76851 7782' 49 




12 


9230 


7793 


5059 457 0979 


5508 


8597 508 0199 518 0270i 48 




13 


410 1833426 0425 


7663 8566 


8071488 1186, 27131 2758' 47 1 




11 


4533 


3058 


442 0278' 6158 473 0634 


3674 5227 5246! 46 




15 


7189 


5637 


28371 8739 


8197 


6212! 7740 7788! 45 




16 


9341 


8318 


5496 453 1325 


5759 


8750 504 0252 519 0219! 44 




17 


411 2492 


427 0949 


8104 3910 


8321 489 1288, 2765, 27t'5! 43 ! 




IS 


5144 


3579:443 0712 6496 474 0882 


3825 5276 5191 


42 ! 




19 


7795 


62081 8319 9080 


3448 


6861 ! 7783 7676 


41 ! 




2) 


412 0445 


8333 5927 459 1665 


6004 


8897 505 0298 520 0161 


40 1 




21 


3096 


428 1467 85341 4248 


8564490 1433i 28u9, 2646 


89 j 




22 


5745 


4)95 444 1140 6S32'475 1124 


8968 5819 5180 


88. 1 




23 


8395 


6723! 3746! 9415 


3683 


6503! 7828 7613 


87 1 




24 


413 1044 


9351 j 6352 460 1993 


6242 


9083 506 0388 521 0096! 36 i 




25 


3693 


429 1979! 8957] 4580 


8801491 15721 2846, 25791 85 i 




26 


6342 


4606445 1562; 7162 476 1359| 4105! 5355! 50611 84 • 




27 


8093 


7233! 4167' 9744 


8917 6638! 7363! 7543 88 




28 


414 1633! 9359! 6771 481 2325 


6474 9171 507 0370 522 0024J 82 1 




29 


42851430 2485| 9375: 4906 


9031 492 17041 2877 


2505 


31 




80 


6932! 5111 1446 1978 74861477 1588 


4286 5384 


4986 


80 




31 


9579; 7736! 4581 462 0066 


4144 


6767 7890 


7466 


29 




32 


415 2226431 0361 


7184: 2646 


6700 


9298 508 0896 


9945 


28 




33 


4872! 2986 


978a! ^'-^25 


9255 493 18291 2901523 2424 


27 ' 




34 


7517! 5810 


447 2388 7804!478 1810! 4859! 5406! 4903 


26 




35 


416 0163| 8234 4990 463 0382 


4364 68891 7910 7381 


25 




36 


2S08432 0857 75911 2960 


6919, 9419 509 0414! 9859! 24 ! 


1 


37 


5453 8481 448 0192: 5588 


9472 494 1948 


2918 524 23861 23 i 




38 


8097 6103 2792 8115 


479 20261 4476 


5421 


4813 


22 




39 


417 0741 8726 5392 464 0692 


4579 1 7005 


7924 


7290 


21 




40 


3385 433 1348 7992 8269 


7131 1 9582 510 0426 


9766 


20 




41 


6028 3970 449 0591! 5345 


9683 495 2060 


2923 525 2241 


19 i 




42 


8871 65911 3190: 8420 


480 2235 


4587 


6429| 4717 


18 i 




43 


418 1313 9212 5789 465 0996 


4786 


7113 


7930 7191 


17 1 




44 


3956:434 1832 8337, 3571 


7337 


9639 511 04311 9665 


16 \ 




45 


6597| 4453 450 0934 6145 


9383 496 2165 


2931 526 2189 


15 1 




46 


9239- 7072- 3582 8719 


481 2488 4690 


5481 


4613 


14 ! 




47 


419 183,)' 9592 6179 466 1293 


4987 7215 


7930 


7085 


13 




43 


4521 435 231l| 8775 8866 


7587 9740512 0429 


9558 


12 




49 


7161 4930 451 1372' 6489 


432 0086 497 2264 


2927 527 2()3t) 


11 




50 


98)11 7543 89671 9012 


2634 


4787 


5425 


4502 


10 


1 


51 


420 2441436 0166, 6583 467 1584 


5182 


7310 


7923 


6973 


9 


1 


52 


50831 2134' 9158, 4156 


7730 


9838 513 0420 


944^^ 


8 


1 


53 


77191 5401 452 1753 6727 


483 0277 493 2355! 2916 528 1914 


7 




54 


421 0353 8018 4347 9298 


2824 


43771 5418 


4838 


6 




55 


2996,437 0634 6941 468 1869 


5370 


7399' 7908 


6358 


5 




56 


5834' 8251! 9535 4489 


7916 


9920 514 0404 


9822 


4 




57 


8272! 5866 453 21231 7009 


484 0462 499 2441 


2899 529 1790 


3 




58 


422 0909! 8432 472l! 9578 


80071 4961 


5393 


4258 


2 




59 


8546 438 1097 7313 469 2147 


5552! 7481 


7887 


6726 


1 




60 


6183 8711 9935, 4716 


8096 500 0000 515 0881 


9198 







' 


65° 64° 63° 1 62° 


61° i 60° 1 59° 


58° 


/ 







Natural Cosines. 






■ 









Natural [ 


Pangeni 


'S. 




465 


' 24° 


25° 


26° 


2r 


28° 


29° 


80° 31° 


/ 





445 2287 466 8077 487 7326 509 5254 531 7094 554 3091577 85C3 600 8606^ 60 


1 


5773! 6618 488 C927i 8919 532 C826| 6894! 73S2 601 2566 59 


2 


9260467 0161J 4530 510 2585' 4559 555 0698 578 1262 0527 


58 


3 


446 27471 37o5| 8133; 6252^ 8293 45041 5144 602 0490, 


57 


4 


6236' 7250 489 17371 ^919 533 2029 831l| 9 )27i 4454 56 


5 


9726 468 07961 5343 51135881 5765 556 2119 579 2912! 8419 55 


6 


447 3216! 43421 8949' 72591 95031 5929! 6797 603 2386 54 


7 


6708 7891J490 2557 512 0930 534 82421 9789 580 €684| 6354| 53 


8. 


448 0200 469 1439! 6166J 46021 6981557 8551 1 4573 604 0323 52 


9 


3693; 4988! 9775! 8275 535 07231 7364! 8462; 4294 51 


10 


71871 8539 491 8888 513 19501 4465 558 1179 581 2353! 8266' 50 


11 


449 0682 470 2090! 0997 5625! 8208 4994| 6245 605 2240| 49 


12 


4178J 5648 492 0610 9802 536 1953 8811582 0189! 6215! 48 


13 


7675! 91961 4224 514 29801 5699 559 2629 4084 606 0192 47 


14 


450 1173 4712751 7888| 665S' 9446| 6449 7980 4170l 46 


15 


4672, 6306 493 1454 515 0838 587 8194 560 0269 583 1828 8149,45 


16 


817l| 9863! 5071 4019 6943| 4( 91 5726 607 2180 44 


17 


451 1672 472 3420 8689 7702 538 06941 7914 9627| 6112 43 


18 


5173 6978 494 2308 516 18S5| 4445 581 ^738 584 8528 608 0095 42 


19 


8876 473 0588 5928 5069 8198 5564| 7481 4r80 


41 


20 


452 2179, 4098 9549 8755 589 1952 9891585 1835 8Q67 


40 


21 


5688' 7659 495 3171517 2441 5707,562 8219, 5241609 2054 89 
9188 474 1222; 67941 0129 9464| 7048! 91481 6043, 38 


22 


23 


453 2694' 4785 496 0418! 9818 540 8221 563 0879,586 3056'610 0034 


37 


24 


6201 1 8849 


4048 518 3508! G9S0: 4710 6965 4026 


36 


25 


9709 475 1914 


7669| 7199 5410740! 8543 587 0876 8019 


35 


26 


454 3218; 5481 


497 1297 519 0891 4501 564 2378i 4788 611 2014 


34 


27 


6728! 9048 


4925 4684 8263! 6213: 8702 6011 


33 


28 


455 0288 476 2616 


8554 8278 542 2027 565 0050 588 2616 612 0008 


82 


29 


3750 6185 


498 2185 520 1974 57911 3888 6583 4007 


31 


30 


7263 9755 


5816 5671 9557! 7728 589 0450! 8008 


30 


31 


456 0776 477 8826 


9449 9868 543 8324 566 1568 4869 613 2010 


29 


32 


4290 6899 


499 8082 521 3067| 70921 5410 82S9 6013 


28 


33 


7806478 0472 


6717 6767.544 0862| 9254590 2211 614 0018 


27 


34 


457 13221 4046 500 0852!522 04681 4682:567 3098! 6134 4024 


26 


35 


4889; 7621 


8989 41701 8404i 6944 591 0058 8082 


25 


36 


8357 479 1197 


7627 7874 545 2177 568 0791! 8984 615 2041 


24 


37 


458 1877 4774 


501 1266 523 1578 5951 4639! 7910 6052 


23 


38 


5397 8352 


4906 5284 9727 8488'592 1889 616 0064 


22 


39 


8918 480 1982 


8547 8990 546 3503.569 2889 5768 4077 


21 


40 


459 2489 5512 


502 2189 524 2698] 7281 ! 619ll .9699 8092 


20 


41 


5962 9098 


5882 6407 


547 1080 570 0045 593 3682 617 2108 


19 


42 


9486 481 2675 


9476525 0117 


4840 3899! 7565 6126 


18 


43 


460 3011 6258503 8121 1 8829 


8621 7755594 1501 618 0145 


17 


44 


6587 9842 


6768 7541,548 2404 571 1612! 5487 4166 


16 


45 


461 0068 482 8427 


504 0415 526 1255 


6188 5471 9375 8188 


15 


46 


8591 7014 


4063 4969 


9973 9381 595 8314 610 2211 


14 


47 


7119 483 0601 


7713 8685 


549 3759 572 3192! 7255 6286 


13 


48 


462 0649 4189 505 1363 527 2402 


7547! 70541596 1196 620 0268 


12 


49 


4179 7778 5015 6120 


550 1385 573 0918! 5140 4291 


11 


50 


7710 484 1868 *8868 9839 


5125 47831 9;184 8821) 


10 


51 


463 1243 4959 506 2822 528 356') 


8916 8649 597 3080 621 2851 


9 


62 


4776 8552 59771 7281 bM 2708574 2516! 6978 6883 


8 


53 


8310 485 2145 9688 520 1004| 65 )2| 6885 598 0926 622 0417 


T 


54 


464 1845 5789 507 3290 4727 552 0297 ;575 0255 4877 4452 


6 


55 


5382 93341 6918 8452 4(:93 4126 8828 8488 


5 


56 


8919 486 2981508 0607 530 2178 7890 7999 599 2781623 2527 


4 


57 


465 2457 6528 4267 59J() 553 1688 57C 1878; 6735 6566 


3 


58 


5996,487 0126 7929 9684| 5488| 5748 600 0691624 0607 


2 


59 


9536 


3726 509 1591531 8364| 92881 9625 


4648 4650 


1 


CO 


466 8077 


7326 5254 7094 554 8091 577 8503 


8606 8694 







(>5° 


(>4" 6;r 02" 1 01° 1 CO" 


69° 1 58° 


' 


Natural Cotangents. | 



46G 



Natural Sines. 



' • 32° 1 33" 34° 35° | 36° | 37° [ 38° | 39° | ' 


529 9193 544 6390 559 1929 573 5764 .o7 7S53 601 8150 615 6615 629 8204 60 


1530 1659 8S30| 4340 8147 5SS 62(6 602 0473 8907, 5464 59 


2 1 4125 545 1269 6751574 0529 25'o3 2795 616 119S 7724 58 


3 i 6591 87071 9162 2911 4gl0 5117: 84S9 99S3 57 


4 1 9;i57i 6145 560.1572 5292 72G2 74S9 573(; 630 2242, 56 


5 531 1521' S5S3i 8931 7672 9613 9760 8069 4500i 55 


6 


S9S6546 1020 6390 575 0053 5S9 1964 603 2080 61T 0359 6758 54 


7 


6450, 34561 8798 2432 4314 4400, 26481 9015 58 


8 


8913: 5S92 561 12i;6' 481li 6663, 6719 4936 681 1272 52 


9 532 1376: 8328! 3614 T190l 9!;12 9C8S 7224 8523 51 1 


10 


8S-S9 547 0763 6021' 9568 590 1361604 1856 9511 5784 50 


11 


63i4 3198 8428 5T6 1946 37i 9 8674 618 1798 SCl-9 49 


12 


8763 5032 562(834 4323! 6057 5991 4(84 682 0293:48 


13 533 1224' 8066! 8239', 6700! 84C4 83C8 6370; 2547i 47 


14 ! 


3GS5 543 0499: 5645' 9076 5910750 605 0624 8655- 4800' 46 


15 


6145 2932 8049 577 1452 3096 2940 619 C989 7053 45 


16 


8605 5365 563 0453 3827| 5442' 5255 3224 93(6 44 


17 


534 10651 77971 2357 62( 2 7787' 7570 5507 633 1557' 4S 


18 


3523549(2281 5260 85T6 592 0182 9SS4 779o! 33(9 42 


19 


5982 2Sr9| 7663 578(950 


2476 606 2198 620 0073: 6(^9' 41 


20 


8440 5 90 564 0066, 8323 


4819 4511 23551 8310 


40 


21 


535 0898 7520 2467! 5696 


7163 6824 4636 634 0559 


39 


22 


3355i 9950 4S69i 8009 95C5 9186 C917 28(8 


88 


23 


5812 550 2379 7270 579 0440 593 1847 607 1447^ 9198 5(57 


37 


24 


8268; 48 Ji 9670| 28121 41S9 3758 621 1478 7805 


86 


25 


536 0724' 7236565 2070| 5183| 6530! 6(G9 3757! 9553 


35 


26 


3179! 9663 


4469' 75531 8871 8379 6036 685 1800 


84 


27 


5034551 2=' 91 


GSC8' 9923 594 1211608 0689 8314 4046 


33 


28 


8; 69; 4513 


9207 580 22921 8550, 2998 622 05921 0292 


32 


29 


537 C543 0944 566 1665! 466l! 5889; 5306, 2870| 8587 


81 


80 


2996: 9370 


4C62 7030! ^228 7614 5146 636 0782 


30 


31 


5449 552 1795 


6459' 9897 595 0566' 9922 7423' 3020 


29 


32 


7902; 4220 


8856 581 17G5I 2904 OC 9 2229 9098' 5270 


28 


33 


538 0354; 6645 567 1252 1 4132: 5241! 4535 023 1974! 7513 


27' 


i 84 


2Sf:6! 9'-C9 3648 649S' 7577i 6S41 42481 9756 


26 


35 


5-257 553 1492 6C43| 63C4' 99131 9147| 6522 637 1998 


25 


! 36 


7708 £915 8437 582 1230 596 2249 610 1452 8796! 4240 


24 


1 S7 


539 (;158 6338 568(8:32! 85951 4cS4; 8756 624 1(;C9| 6481 


23 


' 38 


26:;8 8760 


8225! 5959: C918 6060i 8842: 8721 


22 


1 39 


5; -58 554 11S2 


6619; 8323^ 9252' 8303: 5614638 0901 


21 


! 40 


7507! 36C3 


8011583 0687 597 1586 611 0666i 78851 82(1 


20 


41 


9955 6024 509 0408 


8(50 


8919; 2969 625 01561 5440 


19 


42 


540 24(3 8444 


2795 


5412 


6251 


5270 


2427i 7678 


18 


43 


4851 555 0864 


5187 


7774 


8583 


7572 


40961 9916 
0906 689 2153 


17 


44 


7298 


8283 


7577 584 0136598 (915 


9373 


16 


45 


9745 


57(2 


99C8i 24971 8246 612 2173! 9235: 4890 


15 


46 


541 2191 


8121:570 2357! 48571 5577i 4473 626 15031 6626 


14 


47 


4637 556 0539 


4747 1 72171 79(6 6772, 8771 1 8862 


13 


48 


7(82 


2956 


7186 95771599 0236' 9071i 6038640 1097 


12 


49 


9527 


5373 


0524585 1936 


2565 613 1369! 83(5! 3332 


11 


50 


542 1971 


779: 


571 1912 


4294 


4893^ 8666 627 0571' 5566 


10 


51 


4415 557 0216 


4299 


6652 


, 7221 i 59041 2887| 7799 


9 


52 


6S59 


2621 


6686 


9010 


9549; 82601 51(2 641 0032 


8 


53 


93:'2 


5036 


9073 


586 1367 


600 1876614 0556J 7306! 2204 


T 


54 


543 1744 


7451 


572 14.: 9; 3724 


42(2 


2S52| 96311 4496 


6 


1 55 


4187 


98G5 


8844| 6(;80 


6528 


. 5147628 1894! 6728 


5 


56 


6628 


558 2279 


6229! 8435 


8854 


7442' 4157! 8958 


4 


57 


9069 


4692 


8614,587 0790 


601 1179 


9736 6420:642 1159 


3 


58 


544 1510 


Tl(,5 


573 ( 99S 


3145 


8503 615 2(;29i 86821 3418 


2 


59 


8951 


9517 


3381 


5499 


58271 4822 629 0943 


5647 


1 


60 


G890 


559 1929 


5764 


7853 


8150 66151 3204 


7876 





/ 


57" 


oO)^ 


55° ; 54= 


or r^r 1 51° 


50° 


' 


Natueal Cosines. 



J 



Natural Tangents. 



467 



G 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
23 
21 
22 
23 
24 
25 
25 
27 
28 
29 
3D 
81 
32 
38 
34 
35 
36 
37 
38 
39 
4} 
41 
42 
43 
44 
45 
46 
47 
48 
49 
53 
51 
52 
53 
54 
55 
56 
57 
58 
59 



P.o,^ 



33 



34" 



;24 8394:649 4j76;674 5:38-) 
025 27:^9! 82121 9318 

6783 653 2353 675 3553 
626 0834' 64901 7793 

4SS4651 0631 676 2028 



85' 



3G° 



I 



37° 



88" 



39" 



I 



700 2375 726 5425 753 5541 781 
641ll 9871754 0102 

701 0749 727 4318' 4663 782 
5089! 87671 9232 
9430 723 3218 755 3799 783 

8935' 4774' 6268 702 3773' 7671 j 8339 
627 29S8' 8918 677 05 j9| 8118 729 2125 756 2941 784 
i 7042 652 30641 4752 703 2464' 6582J 7514 
'628 1098 7211! 8997| 6313 730 1041 757 2u9o 785 
! 5155 653 1363 678 3243 704 11631 55011 6668 

92141 5511! 74921 5515! 9933 758 1248 
'629 32741 9363 679 174l| 9839 731 4423| 5829 786 
i 7336'654 38171 5993 705 4224' 8894 759 0413 
1630 1399| 7972 683 0243| 8581 732 33621 4999 787 
I 5464 655 2129, 4531 706 294'3| 7832' 9587^ 

95331 62871 8758' 7331733 23;)3 760 4177 788 

631 3593 656 0447 681 3316 707 1664' 6777| 8769 

I 76371 4639 7276' 602S734 1253 761 3363 789 

632 1733 8772 632 1537 708 0395! 5730| 7959 

I 5310 657 2937. 53 >l\ 4763 735 0210 762^2557 790 

9333; 7103 633 0066' 9133 439l| 7157 
'633 3959 658 12711 4333 709 3534| 9174 763 1759 791 
I 83351 54411 88311 7873 736 3660| 6363 

634 2113 9312 634 2371710 2253! 8147,764 0969 792 
! 6193 659 3785 7143! 663) 737 2336| 5577; 
'635 0274! 7933 635 1416 711 1009! 7127,765 0188 793 

4357 663 2133' 5392! 5390 738 1620 4830' 
8141 6313' 99691 9772i 6115! 9414 794 

635 2527'661 0492633 4247i712 4157,739 0611:766 4031i 
6314 4673i 8528' 8543 5110 8649 

637 0703| 8353 637 281o'713 2931 9611767 3270 795 
4793,662 3043! 7093! 7320740 4113! 7893 
8335 7225 683 1379 714 1712' 8318 768 2517 796 

638 2978 633 1413| 5366! 61U6 741 3124' 7144 
7073 563l| 9955,715 050l! 7633 769 1773 797 

639 1169 9792 639 4246 489S 742 2143! 6404 
5237 664 39341 8533| 9297! 6655 770 1037 798 
9363 8173 693 2332 716 3303 743 1170! 5672 

640 3467 635 2373| 71231 8l00| 5636 771 0309 799 
7569 6570 691 1425 717 2535 744 0204! 4948 

641 1673 663 07691 5725! 6911 1 4724! 95S9 800 
5779 4939 692 0023 718 1319! 9246 772 4233 
9333 91711 4328' 5729 745 377ol 8878 801 

642 3994 667 8374! 8333 719 0141| 8293 773 3523 
8105| 7533693 2939| 4554 746 2324!- 8176 80: 



643 2216 663 1783; 7247| 8970| 
6329 5995 694 1557:723 3337, ( 

644 0444 669 0205' 5838 73 )6[ 



2356 8C9 7840 
7542 810 2658 
22-i9 7478' 
6919 811 2300; 
1611! 7124 
63u5 812 1951' 
10U2' 6780; 
5700813 1611' 
04001 6444' 
51U3 814 1280' 
98U8J 6113 
4515 815 C958 
9224! 5801' 
8935816 0646 
8649! 5493 
3364' 817 0343 
8l82! 5195' 
2302 SIS 0049 
7524J 49{!5 
224S! 9764' 
6975S19 4625 
17031 9483 
6434 820 4354 
11671 9222 
5902 821 4(i93 
0640 8965 
5379 822 3840 
0121! 8718 
4865823 3597 
9311! 8479 
4359 824 3364 
91101 8251' 
3862820 3140, 
8617! 8031' 
3374 826 2925 
8134-1 7821' 
2895 827 2719 
76591 7620 
2425 828 2523 
7193! 7429 
1963829 2337 
6736! 7247 
1511 S30 2160, 



6288 



4560 



7354 774 2327 

47 1S83' 7481803 

6420775 21: 



4417 695 0181:721 2227 748 0956| 6795 834 

8373' 8633| 4493! 665o| 5494 776 1'455, 

645 2797,670 2815| 8813,722 1075 749 0033| 6118 
6918' 7061693 3131 5502| 4575 777 0782 805 

646 104l'671 123)! '^'^'^A ^^'M ^'^^l 5448 
51651 5500 697 1773 723 4361,753 3665 778 0117 806 
9290! 9721| 6397| 8793| 8212| 4788 

647 8417 672 3944 698 0422 724 8227,751 2762| 9460 807 
7546! 8169j 4749| 7663! 7314 779 4135 

648 1676,673 23931 9)78 725 2101 752 lS67i 8812 SOS 



5838 6624699 3409 
994l'674 0854! 7741 
649 4076| 5385 700 2075 
57'^ 56^^ I 55^ 



65401 6423 783 3-192 
726 0932 753 CO 81 1 8173 809 
54251 6541 781 2856 
54° I 5ii°.. I 52° I 5 



y\ 7075 
1067831 1992 
58491 6912 
0632 832 1834 
5418! 6759 
C2i 16 833 1633 
49971 6315 
9790,834 1547, 
458 1! 6481 1 
9382 835 1418! 
41Sli 6357| 
8983 836 1298 
37871 .6242' 
8593 'S37 1138 
341 4 ! 6136' 
8212 838 1187 
8025 I 6041 
T840839 0996 

r 50'' I 



Natural Cotangents. 



468 








Katural Sines 










' 


40° 1 4V 42° 43° 


44° 45° 


46° 


47° 


/ 


042 7876 656 0590 669 13u6 681 9934 694 6584 707 1068 719 3398 731 35371 60 ! 


1643 0104 2785 3468 682 2111 ^ 8676 3124 


5418 5521 1 59 


2 


23321 498J, 5628 4237 695 0767, 5180 


7438 75031 58 : 


8 


4559 7174^ 7789, 6363 


2858, 7236 


9457 9486 57 i 


4 


6785' 9367i 9948 8489 


4949 9291 720 1476 732 1467i 56 


5 


9011657 1560 670 2108 683 0613 


7039 708 1345 3494 


8449 55 1 


6 


644 1236, 3752 4266, 2733 


9128 3393 5511 


5429: 54 ! 


7 


3461 5944 6424' 4861 693 1217] 545l| 7528 


74091 53 i 


8 


5685' 8135 8582' 6984 


3305 75041 9544 


9388: 52 1 


9 


7909 658 0326 671 0739 9107 


5392 9556 721 1559 733 1367i 51 ! 


10 


645 0132- 2516 2895 634 1229 


7479 709 1607 


35741 3345; 50 
5539 5322 49 


11 


23551 4706 50511 3350 


9565, 3657 


12 


• 4577 6895: 7206 5471 697 1651| 5707 


7602 72991 48 


13 


6793! 9083 9361 7591 


3736 '^757 


9615 9275; 47 


14 


9019 659 1271 672 1515' 9711 


5321! 9806 722 1623 734 1250l 46 i 


15 


646 1240 3458 3668 685 1830 


7905 710 1854 


3640 


3225 45 ! 


16 


3460 5645' 58211 8948 


9938: 3901 


5651 


5199 44 ' 


17 


5679 7831 ! 7973 6066 698 20711 5948 


7661 


7173 48 i 


18 


7898 660 0017 673 0125' 8184 4153' 7995 


9671 


9146 42 ' 


19 


647 01161 2202 2276 686 0300 6234 711 0041 


723 1681,735 1118 


41 


20 


2334 4386, 4427 2416 ^315 


2086 


3890 


8090 


40 


21 


4551 6570; 6577 4582 699 0896 


4130 


5693 


5061 


39 


22 


6767 8754' 8727 6647 


2476 


6174 


7705 


7032i 38 i 


23 


8934 661 0936 674 0876' 8761 


4555' 8218 


9712 


9002; 37 i 


24 


648 1199 3119, 3024 687 0375 


6633 712 0260 


724 1719 736 0971 36 


25 


3414 


5300: 5172: 2988 


87111 2303 


3724| 2940 85 


26 


562S 


7482 73191 5101700 0789i 4344 


5729 4908 34 . 


27 


7842 9862' 9466: 7213 28661 6335 


7734 6875! 33 , 


28 


649 0056 662 1842 675 1612' 9325 4942: 8426 


9738 8842 


32 1 


29 


2263' 4022, 3757 688 1435 7018 713 0465;725 1741:737 0808 


31 1 


SO 


4480 6200 5902^ 3546 9093' 2504 


3744 


2773 


30 


81 


6692; 8379' 8046j 5655701 1167| 4543 


5746 


4738 


29 


32 


8903' 663 0557 676 0190 7765 


3241 6581 


7747 


6703 


28 


88 


650 1114 2734! 2333! 9373 


5314! 8618 


9748 


8666! 27 


84 


3324 4910! 4476 639 1931 


7387714 0655726 1748738 0629i 26 


35 


5533 70871 6618 4089i 94591 26911 3748 


2592; 25 


36 


7742 9262 8760 6195 702 1531 


4727 5747 


4553! 24 


87 


9951 664 1437 677 0901 8302 


3601 


6762 7745 


6515! 23 


88 


651 2153 3612' 3041 690 0407 


5672 


8796 9743 


8475! 22 . 


89 


4366 5785J 5181 2512 


7741715 0830:727 1740i739 0435! 21 


40 


6572 7959; 7320 4617 


9811 


2863 


8736 2394! 20 


41 


8778 665 0131! 9459 6721:703 1879 


4895 


5732 4353 


19 


42 


652 0984 


2304 678 15971 8824| 3947 


6927 


7723 6311 


18 


43 


3189 


4475 


3734 691 09271 6014 


8959 


9722 8268 


17 


44 


5394 


6646 


5871 


30291 8081 


716 0939:728 1716 740 0225 


16 ; 


45 


7598: 8817 


8007 


5131:704 0147 


3019 


8710 2181 


15 


46 


9801666 0987,679 0143 


7232 


2213 


5049 


5703 


4137 


14 


47 


653 2004 


3156 


2278 


9332 


4278 


7078 


7695 


6092 


13 


48 


4206 


5325 


4413 692 1432 


6342 


9106 


9686 


8046 


12 


49 


6408 


7493 


6547 


3531 


8406 717 1134 


729 1677 


741 0000 


11 


50 


8609 


9661 


8681 


5630 


705 0469 


3161 


3668 


1953 


10 


51 


654 0810 


667 1828 680 0813 


7728 


2532 


5187 


5657 


8905 


9 


52 


3010 


3994 


2946 


9825 


4594 


7213 


7646 


5857 


8 


53 


5209 


6160 


5078 693 1922 


6655 


9238 


9635 


7808 


T i 


54 


7408 


8326 


7209 4018 


8716:718 1263:730 16231 9758 


6 


55 


9607668 0490 


9:339 6114,706 0776 3287 


3610:742 1708 


5 


56 


655 1804| 2655 


681 1469 8209 2335 5310 


5597 3658 


4 


57 


40021 4818 


3599 694 0304! 4894 7333 


7583 5606 


3 


58 


6193| 6981 


5723 


2398 6953 9355 


9568 7554 


2 1 


59 


8395' 9144 


7856 


4491 9011719 1377 


731 1553 9502 


1 


60 


656 0590 669 1306 


9934 


6584 707 10681 3398 


3537748 1448 





/ 


49° 1 48^ 


47° 


46° 1 45° 1 44° 


43° 1 42° 


' 








N 


ATUBAL 


Cosines. 















Natural Tangents. 




469 





40° 1 41° 1 42° 1 43° 


44° 


45° 


46° 


47° 


t 1 
1 


839 0996 869 2837 900 4040 932 5151 


965 6888 


1-00 00000 


1-03 55303 


1-07 23687 


60 


1 


5955 79781 9309 933 0591 


966 2511 


05819 


61333 


29943 


59 


2 


840 0915 870 3087 901 4580; 6034 


8137 


11642 


67367 


86203 


58 


3 


5878' 82001 9854 934 1479 


967 8767 


17469 


73404 


42467 


57 


4 


841 0844 871 3316 902 51311 6928 


9399 


23298 


79445 


48734 


56 


6 


5812 8435 9 J3 0411935 2380 


968 5085 


29131 


85489 


55006 


55 


6 S42 0782 872 8556| 5693 7834 


969 0674 


34968 


91538 


61282 


54 


7 5755' 8680 934 09T9 936 3292 


6316 


40807 


97589 


67561 


53 1 


8 843 0733 873 3806' 62671 8753 


970 1962 


46651 


1-04 03645 


73845 


52 


9 -67 J8: 8935,905 1557 937 4216 


7610 


52497 


09704 


80132 


51 


10 844 0688 874 40671 685l! 9683 


971 3262 


58348 


15767 


86423 


50 


11 56701 9201906 2147 938 5153 


8917 


64201 


21833 


92718 


4Q 


12 845 0655 875 4338 7446 939 0625 


972 4575 


70058 


27904 


99018 


48 i 


13 5643 9478 907 2748 6101 


973 028(3 


75918 


33977 


1-08 05321 


4T 


14 843 0633 876 4620' 8053 940 1579 


5901 


81782 


40055 


11628 


46 


15| 5625, 9765 908 8360' 7061 


974 156S 


87649 


46136 


17939 


45 


16 847 0620'877 4912 8671941 2545 


724C 


93520 


52221 


24254 


44 


17! 5617 878 0062 909 3934 8033 


975 2914 


99394 


58310 


30573 


48' 


18 843 06171 5215| 9300 942 3523 


8591 


1-01 05272 


64402 


36896 


42 


19 5319 879 0370:910 4919, 9017 


976 4272 


11153 


70498 


43223 


41 


20 849 0624' 552S' 9940 943 4513 


995C 


lt088 


76598 


49554 


40: 


21i 563l|830 0683 911 5265 944 0013 


977 564S 


22925 


82702 


55889 


39: 


22 850 O640 5852 912 0592 5516 


978 133S 


28817 


888C9 


62228 


38 1 


23i 5653 83110171 5922 945 1021 


7027 


34712 


94920 


68571 


87' 


24 851 06671 6186 913 1255 6530 


979 2724 


40610 


1-0501034 


74918 


36: 


25; 5684 882 1357| 6591946 2042 


8424 


46512 


07153 


81269 


351 


26 852 0704| 6531914 1929, 7556 


980 4127 


52418 


13275 


87624 


341 


27i 5723 883 17071 7270 947 3074 


9888 


58826 


19401 


93984 


331 


28 833 0750 6836 915 2815^ 8595 


981 554S 


64239 


25531 


1-09 00347 


32, 


29, 5777,884 2068 7962 948 4119 


982 1256 


70155 


81664 


06714 


31 1 


30 854 0837 7253 916 3312 9646 


6973 


76074 


37801 


18085 


30 i 


31: 5839 8S5 2440I 8865 949 5176 


983 2692 


81997 


43942 


19460 


29! 


32 855 0873 


7630 917 4020 950 0709 


8415 


87923 


50087 


25840 


28' 


33 5910 


836 2S22 9379 6245 


934 4141 


93853 


56235 


32223 


27! 


S4 8>j 0950 


8017 918 4740 951 1784 


9871 


99786 


62388 


88610 


26^ 


35 5992 


887 8215 919 01041 7326 


985 5603 


1-02 05723 


68544 


45002 


25: 


36 857 1037 


84151 5471952 2871 


986 1839 


11664 


74704 


51397 


24 i 


37| 6084 


888 3619 920 0841 1 8420 


7079 


176C8 


80867 


57797 


23' 


38 358 1133 


83251 6214953 8971 


987 2821 


23555 


87035 


64201 


221 


39! 6185 


839 4033,921 1590| 9526 


8567 


29506 


93206 


70609 


21 i 


40 859 1240 


9244| 6969 954 5083 


988 4316 


85461 


99381 


77020 


201 


4l| 6297 


893 4458 922 2350 955 0644 


989 0069 


41419 


1-06 05560 


88436 


19 1 


42 830 1357 


96751 77341 62C8 


5825 


47381 


11742 


89857 


is! 


43| 6419 


891 4894 923 3122 956 1774 


990 1584 


53346 


17929 


96281 


17! 


44 861 1484 


892 01161 85121 7344 


7346 


59315 


24119 


1-10 02709 


16' 


451 6551 


5341:924 3905 957 2917 


991 3112 


65287 


30313 


09141 


151 


46 832 1621 


893 0569 93011 8494 


8881 


71263 


36511 


15578 


I4; 


47i 6694 


5799 925 4700 958 4073 


992 4654 


77243 


42713 


22019 


13 


48 883 1768 


894 1032:923 0102| 9655 


993 0429 


83226 


48918 


28463 


12 


49, 6846 


6268 5506 959 5241 


6208 


89212 


55128 


84912 


ii; 


50 834 1926 


895 1506 927 0914 960 0829 


994 1991 


95203 


61841 


41865 


lOi 


51 7009 


67471 63241 6421 


7777 


1-03 01196 


67558 


47825 


9! 


52 8G5 2i94'893 1991923 1738 961 2016 


995 8566 


on94 


78779 


54284 


8 1 


53 7181J 7238' 7154 7614 


9358 


13195 


80004 


60750 




54 856 2272 897 2487 929 2573 962 8215 


996 5154 


19199 


86233 


67219 


6 


55 7365 


7739| 7996, 8819 997 C953 


25208 


9-2466 


78693 


Si 

|: 

2 


56 867 2460 


898 2994 930 8421963 4427 6756 


31220 


98702 


80171 


57i 7558 


8251 1 8849 964 0037 098 2562 


37235 


1-07 04943 


86658 


58 888 2659 


899 8512 931 4280| 5651 8371 


43254 


11187 


93140 


59 7762 


87751 9714 965 1268 999 4184 


49277 


174851 


99630 


11 

/ 


60 869 2867 


900 4040 932 5151 68881-000 0000 


55303 


23687, 


1-11 06125 


' 1 49" 


48° 1 47° 4(i° 1 45° 


44° 


43° 1 


42° 



NaTUBAL COXANGBNT8. 



40 



470 



Natural Sines. 






48° 


49° 


1 50° 1 51° 


52° j 53' ! 54° ' 




743 1448 


T54 7096 


766 0444 


777 1460 


788 0108 


798 6355 8G9 0170 • 60 


1 


3394 


\ 9004 


1 2314 


i 3290 


1898 


8105 1879 59 




2 


5340 


,755 0911 


4183 


I 5120 


3688 


9355 i 8588 ' 58 




3 


1 72S5 


1 2818 


G051 


1 6949 


5477 


799 1604 6296 57 




4 1 9229 


4724 


7918 


! 8777 


7266 


3352 7004 56 




5 


744 1173 


663^ 


9785 


773 C604 


9054 


5100 8710 ' 55 




6 


3115 


8535 


767 1652 


2431 


789 0841 


6847 810 0416 1 54 




7 


5058 


753 0439 


1 3517 


1 4258 


2627 


8598 2122 53 




8 


6999 


2343 


! 5382 


0084 


4413 


800 0.388 3826 


52 




9 


8941 


42riQ 


i 7246 


'. 7909 


6193 


2088 5530 


51 




10 


745 0881 


6148 


' 9110 


97.33 


7983 


3827 7284 


50 




11 


2321 


8050 


768 0973 779 1557 


9767 


5571 8936 


49 




12 


4760 


9951 


2S85 \ S8S0 .790 1550 


7314 811 0638 i 48 


\ 


13 


6699 


757 1851 


4697 1 5202 


3333 


9056 2339 ■ 47 




14 


8686 


6751 


6558 i 7^24 


5115 


801 0797 4040 j 46 




1 15 ,746 0574 


5650 


S41S = 8345 


6896 


2538 6740 | 45 




16 


2510 


7543 


769 0278 TSO 0665 


8676 


4278 7439 


44 




* 17 


4446 


9446 


2137 ; 2485 791 0456 


6018 9137 


43 




' 18 


6382 


753 1343 


3996 I 4304 ! 2235 


7756 812 0835 


42 




19 


8317 


3240 


5853 ! 6123 ; 4014 


9495 1 2532 


41 




20 747 0251 


5136 


7710 ^ 7940 1 5792 


802 1232 4229 


40 




21 


2184 


7031 


9567 9757 ! T569 


2969 1 5925 


89 




22 


4117 


8926 


770 1423 1%1 1574 i 9345 


4705 i 7620 


38 




23 


6049 


759 0320 


3278 


3390 792 1121 


6440 ' 9314 


87 




24 


7981 


2713 


5132 


5205 


2S96 


8175 818 1008 


36 




25 


9912 


4606 


6986 


7019 


4671 


9909 


2701 


35 




26 


748 1842 


6493 


8340 


6833 


6445 


803 1642 


4398 


34 




27 


3772 


8389 


771 0692 


782 0646 


8218 


8375 


6084 


33 




28 


5701 


760 0280 


2544 


^45^ 


^:?90 


5107 


7775 


32 




29 


7629 


2170 


4395 


4270 


793 1762 


6838 


9460 


31 




30 


9557 


4060 


6246 


6082 


3583 


8569 814 1155 


30 




31 


749 14S4 


5949 


8096 


7892 


5304 


804 0299 i 2844 


29 




32 


3411 


7337 


9945 


9702 


7074 


^028 1 4532 


28 




33 


5337 


9724 


772 1794 


7S3 1511 


8S43 


3756 1 6220 


27 




34 


7262 


761 1611 


3642 


3320 


794 0611 


5484 1 7906 


26 




35 


9187 


3497 


5489 


512T 


2379 


7211 1 9593 


25 




36 


750 1111 


5383 


7336 


6935 , 


4146 


8988 815 1278 


24 




37 


3034 


7263 


9182 


8741 


5913 


805 0664 1 29S3 


23 




38 


4957 


9152 


773 1027 


784 0547 


7678 1 


23S9 1 4^47 


22 




39 


6S79 


762 1036 


2872 ■ 


2352 


9444 i 


4113 : 6330 


21 




40 


8800 


2919 


4716 


4157 


795 1208 ! 


5887 ' 8013 


20 




41 


751 0721 


4802 


6559 


5981 


2972 j 


7560 ; 9695 


19 




42 


2641 


6683 


S4'-i2 


7764 


4735 


9283 SI 6 1376 


18 




43 


4561 


8564 


774 0244 


9566 


6497 


306 1005 3056 


17 




44 


6480 


763 0445 


2086 


7S5 1368 


8259 


2726 4736 


n 




45 


8398 


2325 


8926 


3169 


796 0020 


4446 6416 


15 




46 


752 0316 1 


4204 


5767 


4970 


1780 


6166 8C94 


14 




47 


2233 


6082 


7606 


6770 


3540 


7885 9772 


13 




48 


4149 


7960 


9445 


8569 


6299 ' 


9603 817 1449 


12 




49 


6065 


9838 


775 1288 


786 0367 


7058 


80.7 1321 


3125 


11 




50 


7980 


764 1714 


3121 


2165 


8815 1 


3033 


4801 


10 




51 


9894 


3590 


4957 


3963 


797 0572 ! 


4754 


6476 


^ 




52 


753 1808 


5465 


6794 


5759 1 


2329 ' 


6470 


81.51 


8 




53 


3721 


7340 


8629 


7555 


4i^^S4 


8185 


9324 


7 




54 


5634 


9214 


776 0464 


9350 


5S39 ' 


9899 818 1497 


6 




55 


7546 


765 1087 


2298 787 1145 i 


7594 


308 1612 3169 ' 


5 




56 


9457 


2960 


4132 


2939 


9847 : 


3325 4841 > 


4 




57 


754 1368 1 


4832 


5965 


4732 


793 1100 : 


5037 6512 1 


3 


' j 


58 


3273 1 


6704 


7797 


6524 


2358 ^ 


6749 ! 8182 i 


a^a 


J 


59 


5187 


8574 


9629 


8316 


4604 : 


8460 ' 9852 i 


.^M 


■ 


60 


7096 


766 0444 


777 1460 1 


788 0108 


6855 'i 


509 0170 819 1520 1 


'?^ 


■ 


' 


41° 


40° i 


39° 1 


38-= 


87' 1 3(r i 85° 1 


' 


1 






Natfeal Cosines. 



Natural Tangents. 



471 



/ 


48" 


49° 


50° 


51° 


52° 


53° 
1-32 7044& 


54° 





111 06125;1-15 03684:119 17536!l-23 48972'l-27 99416 


:l-37 63819, 60 


1 


12624 


[ 1044c 


24579! 563191-28 07094 


7848S 


72242 59 


2 


1912; 


1721f 


3162(] 


63672 


14776 


86524 


80672 58 


3 


25635 


23979 


38679 


71030 


22465 


94571 


89108 57 


4 


32146 


30754 


45730 


7839-3 


30160 


1-3302624 


1 97551 56 


5 


38662 


37532 


52799 


85762 


37860 


106841-38 06001 55 


6 


45182 


4431( 


59866i 93136 


45566 


1875C 


j 14458 54 


7 


51706 


51104 


669381-24 00515 


53277 


26822 


' 22922 53 


8 


58235 


57896 


74015 


07900 


60995 


349001 31392 52 


9 


64768 


64693 


i 81097 


15290 


63718 


42984: S9S69 51 


10 


71305 


71495 


83184 


22685 


76447 


51075 


48353 50 


11 


77846 


78301 


95276 


30086 


84182 


59172 


j 56844' 49 


12 


84391 


851121-20 02373 


37492 


91922 


67276 


65342 48 


13 


90941 


91927 


09475 


44903 


99669 


75386 


73847; 47 


14 


97495 


98747 


16581 


52320 


1-29 07421 


83502 


82358 46 


15 


M2 04053 


1-16 05571 


2369-3 


59742 


15179 


91624 


90876 45 


16 


10616 


12400 


30810 


67169 


22943 


99753 


994011 44 


17 


17183 


19234 


37932 


74602 


30713 


1-34 07388 1-39 07934: 43 


18 


23754 


26073 


45058 


82040 


38488 


16029 


16473 42 


19 


30329 


3-2916 


52190 


89484 


46270 


., 24177 


25019 41 


2:) 


36909 


39763 


59327 


96933 


54057 


32331 


33571! 40 


21 


43493 


46615 


6:3468 


1-25 04388 


61850 


40492 


42131! 39 


22 


50081 


53472 


73615 


11848 


69649 


48653 


50698 33 


23 


56674 


60334 


80767 


19313 


77454 


56832 


59272 37 


24 


63271 


67200 


87924 


26784 


85265 


65011 


67352, 36 


25 


69372 


74071 


95085 


84260 


93081 


73198 


76440 ! 35 


26 


76478 


80947 


1-2102252 


41742 


1-30 00904 


81390 


85034 34 


27 


83088 


87827 


G9424 


49229 


08733 


89589 


93636 33 


28 


89702 


94712 


16601 


56721 


16567 


97794 1-4002245, 32 


29 


9u321 


1-17 01601 


23783 


64219 


24407 


1-85 06006 


10360! 31 


30 


1-13 02944 


08496 


3-:970 


71723 


82254 


14224 


19483 30 


31 


09571 


15395 


88162 


79232 


40106 


22449 


28113 29 


32 


16203 


22298 


45359 


86747 


47964 


30680 


36749 28 


33 


22839 


29207 


52562 


94267 


55828 


83918 


45393 27 


34 


29479 


36120 


59769 


1-20 01792 


63699 


47162 


54044 26 


35 


36124 


43038 


G6932 


09323 


71575 


55413 


62702 25 


36 


42773 


49960 


74199 


16360 


79457 


63670 


71367; 24 


37 


49427 


56888 


81422 


24402 


87345 


71934 


80039 23 


38 


56085 


63820 


8S650 


31950 


95239 


80204 


88718 22 


39 


62747 


70756 


95S83 


89503 


1-31 03140 


83481 


97405 21 


40 


69414 


77698 


1-22 03121 


47062 


11046 


9G764 


1-4106093 20 


41 


76086 


84644 


10364 


54626 


18958 


1-36 05054 


14799 19 


42 


82761 


91595 


17613 


62196 


26376 


13350 


23506 18 


43 


S9441 


93551 


24366 


69772 


34801 


21653 


32221 17 


44 


76126 


1-18 05512 


32125 


77353 


42731 


29963 


40943 16 


45 


M4 02815 


12477 


39389 


8494^) 


50668 


88279 


49673 15 


46 


09508 


19447 


46658 


92532 


58610 


48302 


5S409 14 


47 


16206 


26422 


53932 


1-27 00130 


66559 


54931 


67153 13 


48 


22908 


33402 


61211 


07733 


74513 


63267 


75904 12 


49 


29615 


40387 


68493 


15342 


82474 


71610 


84662 11 


50 


36326 


47376 


75780 


22957 


90441 


79959 


93427 10 


51 


43041 


54370 


83081 


30578 


93414 


83315 


1 -42 02200 9 


52 


49762 


61369 


9)331 


88204 


1 -32 06393 


96678 


H!979 8 


53 


56486 


68373 


97687 


45835 


14379 


1-37 05047 


19766 7 


54 


63215 


75382 


1-2304997 


53473 


22370 


13423 


28561! 6 


55 


69949 


82395 


12313 


61116 


80368 


21806 


87362 5 


56 


76687 


89414 


19634 


68765 


38371 


30195 


46171 4 


57 


83429 


96437 


26961 


76419 


46381 


88591 


54988, 3 


58 


9 1176 


1-19 03465 


84-292 


8 1079 


54397 


46994 


63811! 2 


59 


96928 


10493 


41629 


91745 


62420 


55403 


72642 1 


60 


M5 036S4 


17536 


4S972 


99416 


70443 


63S19 


81480; 




41" 


AiV 


81)^ 


r.8" 


1)7° 


:u\ ° 


Ho'^ 1 ' 








Natub 


AL COTA> 


GENTS. 







472 



Natitpal Sines. 



/ 


55° 56° 


57° 


58° 


j 59° 


1 60° 


61° 


' 


819 1520 829 0376 838 6706 848 0481 :857 1673 


866 0254 18-74 6197 1 60 


1 3189 : 2002 : 8290 i 2022 


3171 


1708 1 7607 59 


2 4856 1 3028 : 9873 1 8562 


4668 


3161 I 9016 ' 58 


3 


6523 ! 5252 :889 1455 1 5102 


6164 


4614 ;875 0425 i 57 


4 


! 8189 I 6877 3037 ! 6641 


7660 


6066 ! 1832 i 56 


5 


! 9854 ' 8500 


i 4618 8179 


9155 


7517 3239 i 55 


6 


,820 1519 880 0123 


6199 ! 9717 


,858 0649 


8967 1 4645 1 54 


i 7 


i 8188 1745 • 7778 849 1254 


j 2143 


|867 0417 i 6051 i 53 


8 


; 4846 8366 ' 9:357 2790 


1 8685 


1866 j 7455 i 52 


■. 9 


i 6509 4987 840 0936 4825 


5127 


8814 8859 : 51 


' 10 


1 8170 6607 i 2513 j 5860 


6619 


4762 ;876 0263 l 50 


; 11 


I 9a32 ^ 8226 ' 4090 1 7894 


1 8iC9 


62(9 i 1665 ' 49 


! 12 


821 1492 9845 : 5666 : 6927 


1 9599 


7655 ! 8067 : 48 


13 


3152 881 1463 i 7241 850 0459 


;859 1088 


9100 i 4468 ! 47 


! 14 


4811 3080 ' 8816 


i 1991 


2576 


868 0544 1 5868 : 46 


: 15 


6469 4696 841 0890 


i 8522 


4064 


19S8 ; 7268 i 45 


, 16 


8127 6812 ; 1968 


i 5058 


5551 


8481 ; 8666 ' U 


17 


9784 7927 i 8536 


! 6582 


7(87 


4874 877 0064 '■ 43 


' 18 


822 1440 9541 5108 


8111 


' 8528 


6815 


! 1462 ; 42 


19 


3096 882 1155 6679 


9689 860 0007 


i 7756 


2858 : 41 


20 


4751 2768 . 8249 


851 1167 


1491 


i 9196 


4254 1 40 


21 


6405 438) ; 9819 


1 2698 


2975 


■Sr9 0686 


5649 1 89 


i 22 


8059 5991 842 1888 


4219 


4457 


! 2074 


7C43 88 


; 23 


9712 7602 


2956 


5745 


59f9 


8512 


8437 37 


s 24 


823 1864 9212 


4524 


7269 


7420 


4949 


9880 86 


i 25 


3015 888 0822 


6091 


i 8798 


8901 


f:^'86 


878 1222 


35 


! 26 


4666 24:30 


7657 


'852 0810 


861 0880 


7821 


2613 


84 


1 27 


6816 4038 


9222 


1839 


1859 


9256 


40f:^. 


33 


28 


7965 5646 


8-1-3 0787 


8860 


8887 


870 0691 


5394 


' 82 


29 


9614 7252 


2851 


4881 


4815 


2124 


6788 


1 81 


30 


824 1262 : 8858 


8914 


6402 


6292 


8557 


8171 


30 


31 


2909 834 0463 


5477 


7921 


7768 


4989 


9559 


1 29 


32 


4556 ! 2068 


7089 


9440 


9243 


6420 


879 f946 


; 28 


33 


6202 1 3672 


8600 ,853 C958 


862 0717 


7851 


2882 


27 


34 


7847 : 5275 


844 0161 


2475 


2191 


9281 


8717 


26 


35 


9491 6877 


1720 


8992 


8664 


871 0710 


5ir:2 


25 


36 


825 1185 


8479 


3279 


5508 


5187 


2138 


f48; 


24 


; 37 


2778 


885 0080 


4888 


7023 


6608 


8566 


7869 


23 ! 


38 


4420 


1680 


6895 


• 8538 i 


8079 


4993 


9251 


22 , 


89 


6062 


8279 


7952 


854 0051 ! 


9549 


6419 


880 0683 


21 


40 


7703 


4378 


9508 


1564 863 1019 1 


7844 


2014 i 20 


41 


9343 


6476 


845 1064 


8077 i 2483 \ 


9269 


3894 1 19 


42 


826 0983 


8074 


2618 


4583 1 8956 


B72 0693 


4774 i 18 


43 


2622 


9670 


4172 


6r99 i 5428 1 


2116 


6152 1 17 


44 


4260 


886 1266 


5726 


7609 ! 6889 ' 


858S 


7530 i 16 i 


45 


5897 


2862 


7278 


9119 1 8:355 | 


4960 


8907 i 15 j 


46 


7534 


4456 


8880 


855 0627 1 9820 ■ 


6881 


881 0284 j 14 j 


47 


9170 


6050 


846 0881 1 


21:35 864 1284 ! 


7301 


1660 13 


48 


827 0806 


7648 


1982 ! 


3643 i 


2748 i 


9221 


3035 : 12 


49 


2440 


9286 


8481 i 


5149 i 


4211 > 


573 0640 i 


4409 ; 11 


50 


4074 837 0827 I 


5030 


6655 : 


5678 j 


2058 j 


5782 


10 


51 


5708 


2418 


6579 


8160 j 


7184 1 


3475 


7155 


9 


52 


7840 


4009 


8126 


9664 • 


8595 ; 


4891 1 


8527 ' 


8 


53 


8972 


5593 


9678 


856 1168 865 0055 1 


6807 


9898 ; 


7 


54 


828 0)03 


7187 


847 1219 


2671 ! 


1514 1 


7722 i 


B82 1269 j 


6 


55 


2284 


8775 


2765 


4173 ! 


2973 i 


9187 ! 


2688 ' 


5 


56 


8834 888 0363 


4809 


5674 i 


44:30 i 


>74 0550 ! 


4C07 


4 


57 


5498 


1950 


5853 


7175 1 


5887 


1963 


5876 


3 


58 


7121 


8536 


7397 


8675 


7344 


8375 


6748 


2 


59 


8749 


5121 


8939 


857 0174 1 


8799 


4786 


8110 j 


1 


60 


829 0376 


6706 


848 0481 


1673 \ 


^QQ 0254 


6197 


9476 i 





' 


a4' 


83° 


32° 


81° 1 


80° 


29° 


28° 1 


' 


Natural Cosines. 


( 
1 



Natural Tangents. 



473 



' 


55" 


56° 


57° 


58° 


59° 


60° 


61° 


' 





1-42 81480 


1-48 25610 1-53 98650 1*60 03345 1 66 42795 


1-73 205G8 


1-8040478' 60 


1 


90326 


349161-54 08460 


187C9 


58766 


32149 


52860 


59 


2 


99178 


44231 


18280 


24082 


64748 


43803 


65256 


58 


3 


1-43 08039 


53554 


28108 


84465 


75741 


55468 


77664 


57 


4 


16906 


62884 


37946 


44858 


86744 


67144 


90086 


56 


5 


25781 


72223 


47792 


55260 


97758 


78833 1-81 02521 


55 


6 


34664 


81570 


57647 


65672 


1-6708782 


90588 


14969, 54 


T 


43554 


9G925 


67510 


76(94 


19818 


1-7402245 


274301 53 


8 


52451 


1-49 00288 77383 


86525 


30864 


13969 


89904' 52 


9 


61356 


09659 j 87264 


96966 


41921 


25705 


52391 1 51 


10 


70268 


19039' 97155 1-61 07417 


52988 


37453 


64892 50 


11 


79187 


284261-55 07054 


17873 


64067 


49213 


77405; 49 


12 


88114 


37822 


16963 


28349 


75156 


60984 


89982 48 


13 


97049 


47225 


26880 


88329 


86256 


72768 1-82 02473! 47 


14 


1-44 05991 


56637 


36806 


49820 


97367 


84564 


15026 


46 


15 


14940 


66058 


46741 


59820 


1-6808489 


96371 


27598 


45 


16 


23897 


75486 


56685 


70380 


19621 


1-7508191 


40173 


44 


17 


32362 


84923 


66689 


80850 


30765 


20028 


52767 


43 


18 


41834 


94367 


76601 


91883 


41919 


31866 


65874 


42 


19 


50814 


1-50 03821 


865721-62 01920 


58085 


^ 48722 


77994 


41 


20 


59801 


13282 


96552 


12469 


64261 


65590 


90628 


40 


21 


68796 


22751 


1-56 06542 


23029 


75449 


67470 


1-83 03276 89 


22 


77798 


82229 


16540 


88599 


86647 


79362 


15936 


88 


23 


86808 


41716 


26548 


44178 


97856 


91267 


28610 


37 


24 


95825 


51210 


36564 


54768 


1-69 09077 


1-76 03183 


41297 


36 


25 


1-45 04850 


60713 


46590 


65868 


20308 


15112 


58999 


85 


26 


13883 


70224 


56625 


75977 


81550 


27058 


66713 


84 


27 


22923 


79743 


emed 


86597 


42804 


89007 


79442 


88 


28 


31971 


89271 


76722 


9T227 


54069 


50972 


92184 


82 


29 


41027 


9S807 


86784 


1-63 07867 


65844 


62950 


1-84 04940 


81 


30 


50C90 


1-51 08352 


96856 


18517 


76631 


74940 


17709 


30 


31 


59161 


17935 


1-57 06936 


29177 


87929 


86943 


30492 


29 


32 


68240 


27466 


17026 


89847 


99238 


98958 


482S9 


9q 


33 


77326 


37030 


27126 


5:,528 


1-7010559 


1-7710985 


56099 


27 


34 


86420 


46614 


37234 


61218 


21890 


28024 


6S923 


26 


35 


95522 


56201 


47352 


71919 


38288 


35076 


81761 


25 


36 


1-46 04632 


65796 


57479 


82680 


44587 


47141 


94613 


24 


37 


13749 


75400 


67615 


98851 


55958 


59218 


1-85 07479 




38 


22874 


85012 


77760 


1 -04 04(82 


67329 


718U7 


20358 


22 


39 


32007 


94632 


87915 


148-34 


78717 


884C9 


38252 


21 


40 


41147 


1-52 04261 


98079 


25576 


9:116 


95524 


46159 


2) 


41 


50296 


13899 


1-58 08253 


36338 


1-7101527 


1-7807651 


59080 


19 


42 


59452 


23545 


18436 


47111 


12949 


19790 


72015 


18 


4S 


68616 


332 JO 


28628 


57898 


24882 


31948 


84965 


17 


44 


77788 


42S03 


38830 


68687 


85827 


44107 


97928 


16 


45 


86967 


52535 


49041 


79490 


47283 


5C2S5 


1-86 10905 


15 


46 


96155 


62215 


59261 


90804 


58751 


68475 


28896 


14 


47 


1-47 05350 


71904 


69491 


1-65 01128 


70230 


80678 


3(^902 


13 


48 


14553 


81602 


79731 


11968 


81720 


92893 


49921 


12 


49 


23764 


91308 


89979 


22808 


98222 


1-79 05121 


62955 


11 


50 


32983 


1-53 01028 


1-59 00238 


88668 


1-72 04786 


17362 


76008 


10 


51 


42210 


10746 


10505 


44529 


16261 


29616 


89065 


9 


52 


51445 


2(H79 


20788 


55405 


27797 


41883 


1-87 02141 


8 


53 


60688 


3 219 


31070 


66292 


89346 


54162 


15231 


7 


54 


69938 


39969 


41866 


77189 


50905 


66454 


28336 


6 


55 


79197 


49727 


51672 


88C97 


62477 


78759 


41455 


5 


56 


88463 


59494 


61987 


99016 


74060 


91077 


54588 


4 


57 


97738 


69270 


72812 


106 09945 


85654 


1-80 08408 


67730 


3 


5S 


1-48 07021 


79054 


82647 


21)884 


97260 


15751 


80898 


2 


59 


16311 


88848 


92991 


31884 


1-78 08^78 


28108 


94074 


1 . 


60 


25610 


98650 


1-60 03845 


42795 


2.5„8 


40478 


1-88 07265 





' 


34° 


r>:-{° 


32" 


31° 


30^ 


2r 


28° 


' 



Natuhal Cotangents. 

40* 



474 






Ivatural Sines. 










62° 


C3° 


64° 1 05° 06° 


or 


68° 1 ' 





682 9476 


891 0065 


898 7940 906 3078 913 5455 


920 5049 


927 18i39 


60 


1 


883 0841 


1885 


9215 


4307 


6637 


6185 


2928 


59 


2 


2206 


2705 


899 0489 


5535 


7819 


7320 


4016 


58 


3 


8569 


4024 


1763 


6762 


9001 


8455 


5104 


57 


4 


4938 


5342 


3;86 


7989 914 0181 


9589 


6191 


56 


5 


6295 


6659 


4807 


9215 


1861 


921 0722 


7277 


55 


6 


7656 


7975 


5578 


907 0440 


2540 


1854 


8868 


54 


T 


9017 


9291 


6848 


1665 


3718 


2986 


9447 


58 


8 


884 0377 


S92 0606 


8117 


2888 


4895 


4116 


928 C531 


52 


9 


1736 


1920 


93S6 


4111 


6072 


5246 


1614 


51 


10 


3C95 


8234 


900 0654 


5338 


7247 


6375 


2696 


50 


11 


4453 


4546 


1921 


6554 


8422 


7504 


3778 


49 


12 


5810 


5858 


3188 


7775 


9597 


8632 


4858 


48 


13 


T166 


7169 


4453 


8995 915 0770 


9758 


59S8 


47 


14 


8522 


8480 


5718 90S 0214 


1943 


922 0884 


7017 


46 


15 


9876 


9789 


6982 


1482 


3115 


2010 


8096 


45 


16 


885 1230 


898 1C98 


8246 


2649 


4286 


8184 


9173 


44 


IT 


2584 


2406 


9508 


3866 


5456 


4258 


929 C250 


48 


18 


3986 


3714 


901 0770 


5082 


6626 


5381 


1826 


42 


19 


5288 


5021 


2081 


6297 


7795 


6503 


2401 


41 


20 


6639 


6826 


3292 


7511 


8963 


7624 


3475 


40 


21 


T9S9 


7632 


4551 


8725 916 0130 


8745 


4549 


39 ^ 


22 


9839 


8936 


5810 


9938 


1297 


9865 


5622 


88 


23 


886 0688 


894 0240 


7068 9C9 1150 


2462 


928 C9S4 


6694 


37 


24 


2086 


1542 


8825 


2361 


3627 


2102 


7765 


36 


25 


3388 


2844 


9582 


3572 


4791 


8220 


8835 


85 


26 


4730 


4146 


902 0838 


4781 


5955 


4336 


9905 


34 


27 


6075 


5446 


2C92 


5990 


7118 


5452 


980 C974 


33 


28 


T420 


6746 


3347 


7199 


8279 


6567 


2042 


82 


29 


8765 


8045 


4600 


8406 


9440 


7682 


81(9 


31 


30 


887 0108 


9844 


bSoS 


9618 917 0601 


8795 


4176 


30 


81 


1451 


895 0641 


7105 910 C819 


1760 


9908 


5241 


29 


32 


2793 


1938 


8856 


2024 


2919 


924 1020 


6306 


28 


38 


4134 


3234 


9606 


3228 


4077 


2181 


7370 


27 


81 


5475 


4529 


908 C856 


4432 


5284 


3242 


8434 


26 


35 


6815 


5824 


21(5 


5635 


6391 


4351 


9496 


25 


36 


8154 


7118 


3353 


6887 


7546 


5460 


931 0558 


24 


37 


9492 


8411 


4600 


8038 


8701 


6568 


1619 


23 


38 


888 0830 


9703 


5S47 


9238 


9855 


7676 


2679 


22 


39 


2166 


896 (994 


7093 911 0438 918 10C9 


8782 


8739 


21 • 


40 


3503 


2285 


8838 


1637 


2161 


9888 


4797 


20 - 


41 


4838 


3575 


9582 


2835 


8313 


925 C998 


5855 


19 ■ 


42 


6172 


4864 


904 0825 


4033 


4464 


2097 


6912 


IS * 


43 


75C6 


6158 


2068 


5229 


5614 


8201 


7969 


17 '• 


44 


8839 


7440 


3810 


6425 


6768 


4303 


9024 


16 ^ 


45 


889 0171 


8727 


4551 


7620 


7912 


5405 


932 0079 


15 t^ 


46 


15C3 


897 0014 


5792 


8815 


9060 


6506 


1138 


1* : 


47 


2834 


1299 


7032 912 0008 919 0207 


7606 


2186 


18 fe 


48 


4164 


25S4 


8271 


1201 


1353 


8706 


8288 


12 fe 


49 


5493 


3868 


9509 


2398 


2499 


9805 


4290 


11 


50 


6822 


5151 


905 0746 


3584 


3644 


926 (.902 


5340 


10 


; 51 


8149 


6483 


1983 


4775 


4788 


2000 


6390 


9 


i 52 


9476 


7715 


3219 


5965 


5931 


3096 


7439 


8 


1 53 


890 0808 


8996 


4454 


7154 


7078 


4192 


8488 


7 


i 54 


2128 


893 0276 


5688 


8842 


8215 


5286 


9535 


6 


1 55 


8453 


1555 


6922 


9529 


9356 


6380 


933 C5S2 


5 


56 


4777 


2834 


8154 913 0716 920 0496 


7474 


1628 


4 


5T 


6100 


4112 


9386 


1902 


1635 


8566 


2678 


8 


58 


7428 


5389 


906 0618 


8087 


2774 


9658 


3718 


2 


69 


8744 


G665 


1848 


4271 


3912 


927 0748 


4761 1 


1 


60 


891 C065 


7940 


3078 


5455 


5049 


1839 


5804 : 





; 


27° 


2G' 


25° 


24' 


9:^ 


2'1° 


21° 1 


' 








NATt 


rcAL Cosr 


sm. 









Xattjral Tangents. 



'1 02° 1 03° 64° 


05° 


00° 


or 08° 


' 





1-88 07265 1-96 26105 2-05 08038 2-1445069 2-24 60868 2-85 585242-47 50869! 60 


1 


2047C 


40227 


18185! 61866; 77962 7759;) 71612 


159 


2 


88690 


54864 


83849, 77683 95580| 966S8 93886 


1 ^8 


3 


46924 


68518 


48581 1 94021 2-25 18*221 2-86 15801 2-48 18190 


57 


4 


60172 


82688 


63782 2-15 103781 30885J 84946 34028 


56 


5 


78486 


96874 


78950 


26757 


48572 54118 54887 


55 


6 


837181-9711077 


94187 


48156 


66288 73816! 75781 


54 


T 


1-89 00006 


25296 2-06 09442 


59575 


84016 92540 96706 


53 


8 


13818 


89531 


24716 


76015 2-26 01778 2-87 11791 2-49 17660 


52 


9 


26685 


58782 


40008' 92476 


19554 


81068, 88645 


51 


10 


89971 


68050 


55318 2-16 08958 


87857 


50872 59661 


50 


11 


53822 


82884 


70646, 25460 


55184 


69708 80707 


49 


12 


66688 


93685 


85994 41983 


73035 


89360 2-50 01784 


48 


13 


80068 1-98 10952'2'07 01859' 58527 


90909 2-3808444 22891 


47 


14 


98464 


25-286 


16748 750912-27 08807 


27855 44029 


46 


15 


1-93 06874 


89686 


821461 91677 


26729 


47293, 65198 


45 


16 


20299 


54003 


47567 2-17 08288 


44674 


66758 86398 


44 


17 


88788 


08887 


68007 


24011 


62643 


86250 2-5107629 


43 


18 


47198 


82787 


78465 


41559 


83686 2-39 05769 28893 


42 


1 19 


60668 


97204 


93942 


58229 


98658 


- 25316 50183 


41 


20 


74147 


1-9911687 2-08 09488 


74920 2-2816698 


448S9 71507 


40 


i 21 


87647 


26087 


24958 


916811 84758 


644931 92868 


39 


! 22 


1-9101162 


40554 


40487 2-18 08864! 52846 


84118 2-52 14249 


88 


i 23 


14691 


55038 


56089 


25119 70959 2-40 08774 85667 


87 


1 24 


2S286 


69589 


71610 


41894 89096 


28457 57117 


86 


25 


41795 


84056 


87200 


58691 2-29 07257 


48168 7859S 


85 


26 


55870 


9859J 2-09 02809 


75510 


25442 


62936 2-53 00111 


84 1 


27 


68980 2-00 18142 


18487 


92349 


48651 


82672 21655 


33 1 


28 


32565 


27710 


34085 2-19 09210 


61885 2-4102465! 43231 


32 


29 


96186 


42295 


49751 


26098 


83148 


22286 64839 


81 1 


30 


1-920982: 


56897 


65436 


42997 


98425 


42186' 86479 


80 i 


31 


28472 


71516 


81140 


599-28 2 8016782 


62018 2-5408151 


29 1 


82 


37188 


86158 


96864 


76871 


35364 


81918, 29S55 


28 1 


33 


50819 2-01 00S06'2-1C 12607 


98840 


58423 


2-42 01851 51591 


27 


34 


64516 


15477 2S869 2-23 10881 


71801 


21812, 73359, 26 | 


35 


78228 


30164 4415;) 


27848 


93206 


4183ll 95160 25 


36 


91956 


44839 59951 


44878 2-8108637 


61819 2-5516992 24 ; 


37 


1-9305699 


59592 75771 


61934 


27(.92 


81864 33858! 28 1 


88 


19457 


74;331 91611 


79312 


451712-4301938 60756122 1 


39 


38281 


89088 2 11 07470 


98112 


64076 


220411 82886! 21 


40 


4702 J 2-02 08S62| 28348 2-21 18284 


82306 


42172 2-5604649 2) 


41 


60825 186541 89246 


30379 2-32 01163 


62831, 26645! 19 


42 


74645 


88462, 55164 


47545 


19740 


82519 4S674 


18 i 


48 


88481 


48-289 71101 


64738 


38845 2-44027361 70735 


17 ^ ! 


44 


1-9402833 


681881 87057 


81944 


56975 


229S2 92830 


16 * 


45 


16200 


77994;2 12 08084 


99177 


75630 


48256 2-57 14957 


15 ; 


46 


80088 


92873 


19333 2-2216432 


94311 


68559, 87118 


14 ! 


47 


48981 2-03 07769 


85046 


83709 2-8818017 


88891 1 59812 


13 j 


48 


57896 


22688 


51082 


51009 


81748 2-45 04252 81589 


12 : 


, 49 


71826 


37615 


67137 


68331 


50505 


24642 2-58 03800 


11 i 


50 


85772 


52565 


88218 


85676 


69287 


45061; 26094 


10 


' 51 


99788 


67582 


99308 2-23 08048 


88095 


65510 48421 


9 ! 


52 


1-9518711 


S2517 213 15428 


23488 2-84 06928' 85987 1 70782 


8 


53 


27704 


97519, 31559 


37845 


25787 2-46 06494 98177 


7 


1 54 


41718 2-0412540 


47714 


55230 


44672 27080 2-5915606 


6 


! 55 


55789; 27578 


68890 


72788 


68582 47596, 83068 


5 


1 56 


69780 


42684 


80085 


90218 


8-2519 68191 60564 


4 


i 57 


83887 


57708 


938i)l'2-24 07721 2-35014811 88816' 88095 


3 


i 58 


97910 


72800 2-14 12587 


25247 


20469 2-47 09470 2-6005659 


2 


59 


1-9612000 


87910 


28798 


42796 


89488 


30155 28258 


1 


60 


26105 2-0508088 


45069 


60868 


58524 


50869 60891 





' 


27° ' 20° 


25° 


24° 


28° 


22° 21° ' 1 




Natural Cotangents. | 



476 



Natural Sini^s. 



69° 


70° 


1 71° 1 72° 


73° 


74° 


1 75° 


r 


T 





983 5804 


939 6926 


945 5186 951 0565 


956 3048 


961 2617 


965 9258 


60 




1 


6846 


i 7921 


j 6132 1 1464 


3893 


8418 


966 0011 


59 




2 


7888 


1 8914 


7078 2861 


4747 


4219 


0762 


58 




8 


8928 


1 99J7 


8023 3258 


5595 


5019 


1518 


57 




4 


9968 


940 (899 


8968 4154 


6443 


5818 


2268 


56 




5 


934 1007 


1891 


9911 1 5050 


! 7290 


6616 


3012 


55 




6 


2045 


2831 


946 0854 1 5944 i 8136 


7413 


3761 


54 




T 


8082 


8871 


1795 6S88 i 8981 


8210 


45G8 


53 




8 


4119 


4860 


2736 7781 | 9825 


9005 


5255 


52 




9 


5154 


5843 


3677 8623 957 0609 


9800 


6001 


51 




10 


6189 


6885 


4616 I 9514 


1512 


962 0594 


6746 


50 




11 


7223 


7822 


5555 952 0404 


2354 


1387 


7490 


49 : 




12 


8257 


8808 


6493 j 1294 


8195 


2180 


8234 


48 




13 


9289 


9793 


7430 1 2183 


4035 


i 2972 


8977 


47 




14 


935 0321 


941 0777 


8366 


1 8071 


4875 


; 8762 


9718 


46 




15 


1352 


1760 


9301 


1 8958 


5714 


\ 4552 


967 0459 


45 




16 


2382 


2743 


947 0286 


' 4844 


6552 


5342 


1200 


44 




1 IT 


8412 


3724 


1170 


5780 


7389 


6130 


1939 


43 




18 


4440 


4705 


2103 


6615 


8225 


6917 


2678 


42 




19 


5468 


5686 


3085 


7499 


9.:)60 


7704 


8415 


41 ^ 


£ 


1 20 


6495 


6665 


8966 


8382 


9895 


8490 


4152 


40 1 


1 


1 21 


7521 


7644 


4897 


9264 


958 0729 


9275 


4888 


39 ? 


f 


1 22 


8547 


8821 


5827 


953 0146 


1562 


963 0060 


5624 


88 




23 


9571 


9598 


6756 


1027 


2394 


0843 


6358 


37 




24 


936 0595 


942 0575 


7684 


1907 


3226 


1626 


7092 


36 




25 


1618 


1550 


8612 


2786 


4056 


2408 


7825 


35 




26 


2041 


2525 


9533 


8664 


4836 


8189 


8557 


34 




27 


8662 


8493 


948 0464 


4542 


5715 


8969 


9283 


83 




28 


4683 


4471 


1389 


5418 


6543 


4748 


96S 0018 


82 




29 


57u3 


5444 


2313 


6294 


7371 


5527 


0748 


81 




30 


6722 


6415 


3237 


7170 


8197 


6805 


1476 


30 




31 


7740 


7386 


4159 


8044 


9023 


7081 


2204 


29 


82 


8758 


8355 


5081 


8917 


9848 


7858 


2931 


28 


83 


9774 


9324 


6002 


0790 


959 0672 


8683 


8658 


27 


34 


987 0790 


943 0293 


6922 


954 0662 


1496 


9407 


4383 


26 


85 


1806 


126^ 


7842 


1533 


2318 


964 0181 


5108 


25 i. 


86 


2820 


2227 


8760 


2408 


8140 


C954 


5832 


24 J| 


87 


8833 


3192 


9678 


8273 


8961 


1726 


6555 


23 H 


88 


4846 


4157 


949 0595 


4141 


4781 


2497 


7277 


22 M 


89 


5858 


5122 


1511 


50&9 


5600 


3268 


7998 


21 m 


40 


6869 


6085 


2426 


5876 


6418 


4037 


8719 


20 M 


41 


T8S0 


7048 


8341 


6743 


7236 


4806 


9438 


19 ^P 


42 


8889 


8010 


4255 


7608 


8058 


5574 


969 0157 


18 fe. 


1 


43 


9898 


8971 


5168 


8473 


8869 


6841 


0875 


17 *; 


1 


44 


938 C9J6 


9931 


6080 


9336 


9684 


7108 


1593 


16 t 


1 


45 


1913 


944 C890 


C991 


955 0199 


960 0499 


7873 


2309 


15 H 


1 


46 


2920 


1849 


79:52 


1062 


1312 


8638 


3025 


14 .' 


1 


47 


8925 


2807 


8812 


1923 


2125 


9402 


3740 


13 




48 


4930 


8764 


9721 


2784 


2937 


965 0165 


4453 


12 




49 


5934 


4720 


050 0629 


864:3 


3748 


0927 


5167 


11 




50 


6988 


5675 


1536 


4502 


4558 


1689 


5879 


10 




51 


7940 


6630 


2443 


5861 


5363 


2449 


6591 


9 




52 


8942 


7584 


3348 


6218 


6177 


8209 


7301 


8 




53 


9943 


8587 


4253 


7074 


6984 


3963 


8011 


7 




54 


939 C948 


9489 


5157 


7930 


7792 


4726 


8720 


6 




55 


1942 


945 0441 


6061 


8785 


8598 


5484 


9428 


5 




56 


2940 


1391 


6963 


9639 


9408 


6240 


970 0186 


4 




57 


8938 


2341 


7865 


956 0492 961 0208 1 


6996 


0842 1 


3 




58 


4935 


8290 


8766 


1345 


1012 


7751 


1548 i 


2 




59 


5931 


4238 


9666 


2197 


1815 


8505 


2253 


1 




60 


6926 


5186 


951 0565 


3048 


2617 


9358 


2957 







' 


2(r 


]V)" 


IS^ 


17° 


16° 


15° 


14° 


1 1 




NATtiRAL Cosines. i 





jS^atural Tangents. 



477 



' 


69° 70° 71° 72° 73° 


74° 75° 


' 


! « 


2-60 50891 ;2-74 74774 290 42109 8-07 76885 3-27 08526 3-48 74144'8-73 20508 60 


1 


73558 996611 69576 3-08 07825 42588 8-4912470 63980 59 


2 


96259 2-7524588 970S9 87869 76715, 50874 3-74 07546 58 


8 


2-6118995; 49554 2-9124649; 68468 3-28 If 907 89356 51207 57 


4 


4176( 


) 74561 52256 99122, 45164 3-50 27916: 94963 56 


5 


64571 


99608 799(9 8-09 298311 79487^ 66555' 3-75 88815 55 


6 


87411 


2-76 24695 2-92 07610J 60596 3-29 18876 8-51 052781 82768 54 


7 


2-6210286 


49822, 85858 91416 48330; 44070 3-76 26807 58 


S 


33196 


74990 68152 3-10 22291' 82S5ll 82946' 7(947 52 


9 


561412-77 00199 9r995 58223 8-30 17488 3-52 21902 8-77 151 85 51 


10 


79121 25448 2-9818885 84210, 52091 6< 938i 59519 50 


11 


2-6802136 50788, 46822 8-1115254' 868113-53 00054 8-78 03951 49 


12 


25186 76069 74807i 46853 3-81 21598 892511 4848148 


13 


48271,2-78 01440 2-9402840' 775^9 56452 785281 9S1C9 47 


14 


71392 


2685£ 


30921 3-12 08722; 91878 3-54 17886 3-79 87835 46 


15 


94549 


52807 


59:150, 899913-32 26362 57825! 8266145 


16 


2-64 17741 


77802 


87227i 718171 61419 96846 3-80 27585 44 


17 


40969 2-79 03389 2-95 15453 3-13 027011 96548 3-55 36449' 72609 48 


18 


64282! 289171 48727| 34141 3-33 31786 76138 3-8117738 42 


19 


87581 '545871 72050 65639 66^7 3-56 15900| 62957 41 


20 


2-6510867! 80198 2-96 00422 97194 8-34 02326 55749 8-82 08281 40 


21 


34238 2-80 05901 


28842 3-14 288071 37724 95681 53707, S9 


22 


57645 81646 


57312 604781 73191 3-57 35696 99238 88 


28 


81089 57483 


85831' 92207 3-35 08728 75794 3-88 4486187 


24 


2-66 04569 83263 2-97 14399 8-15 23994! 44883 3-58 1^975| 90591 86 


25 


28085 2-81091341 48016 55S40! 80008 562413-84 36424 35 


26 


51688 


850481 71688 87744 3-3615758 96590l 82358 84 


27 


75227 


61004 2-93 00400 8-16 19706, 51568 3 59 87024 3-85 28396 38 


j 28 


98858 


87003 


29167| 51728' 87458 77543| 74587i 32 


! 29 


2-67 2-2516 2-82 13045 


57983 888; 8 3-37 234( 8 3-60 18146 3-86 20782 81 


i 80 


46215 


89129 


86850 8-17159481 59434 58835! 67131, 80 


1 31 


69951 


65256 2-9915766 48147| 95531' 996f 9 3-87 18584' 29 


! 82 


93725 


914261 44734' 80406 3-88 31699 8-61 404691 60142 28 


1 88 


2-6817585 2-88176891 737518-18 12724! 67938 81415 3-88 06805 27 


84 


418881 43896 3-00 02820i 45102 8-89 04249 3-62 22447] 58574 26 r 


35 


65267' 70196 


31939 77540 40631 63566 3-89 00448 25 1 


86 


89190| 96539 


61109 3-1910039, 77085 3-68 047711 47429 24 1 


87 


2-69 18149 2-84 22926 


90830; 42598 3-40 13612 46064 94516 23 


38 


87147 49356 8-0119308 752171 50210' 87444 3-9041710. 22 ! 


89 


61181 758311 48926 3-20 07fe97| 86882 3-64 2891 1| 89011 21 


40 


85254 2-85 02349 7830 1| 40638 3-41 23626 70467,3 91 36420l 20 


41 


2-70 09364 289118-0207728 78440| 60443 3-651 21 ll| 83987, 19 


42 


33518 55517 


87207 3-21 063041 97388 53844 3-92 81568 18 


43 


57699 82168 


667871 89228 3-42 34297, 95665| 79297| 17 


44 


81928 2-86 08868 


968201 72215! 71384 3-66 87575,3 98 27141 16 


45 2-71061861 


35602 3-03 25954 3-22 05263 3-43 08446 79575 75094 15 


46 


30487 


62386 556411 88873, 45631 3-67 21665 3-94 231 57 14 


47 


54826 


89215 853811 71546' 82S91 638451 71831 13 


48 


79204 2-87 16u88 3-04 15173 3-23 04780 3-44 2f 220 8-6806115 8-95 19615 12 1 


49 2-72 036201 4S007| 45018 88178! 57685 4S475I 68011 11 | 


50 


28076 69970| 74915 71438 95120 90927 8-9616518 10 


51 


52569! 96979 3-05 04866 3-24 04860 8-45 32679 3-69 884691 65137. 9 


52 


77102 2-88240331 


34870, 88346 70315 76104 3 9713868 8 


53 2-78 016741 51182| 


649-28 71895 3-46 08026 8-7018830! 62712 7 


54 


262841 782771 


95038 3-2505508 45813 61 64S 3-9811669 6 


55 


50934 2-89 05467 8-06 25203, 89184' 83676 8-7104558 60739 5 | 


56 


75623! 


82704; 554211 72924 3-47 21616 475613-99 09924 4 


67 2-74 008521 


5998) 85694 3-26 06728 59632 90658 59223 8 


58 


251201 


87314 3-07 1602)1 40596 97726 3-72 83847 4-00 08636 2 


59 


49927 2-9014688! 46400| 74529 3-4885896 77181 1 58165 1 


60 


74774 42109 76835 8-2708526, 74144 8-73 205084-0107809 


' 


20° 19° 1 18° 1 17° 1()° 1 15° 1 14° 1 ■ 


Natural Cotangents. 



4:1^ 



Natural Sixes. 



1 / 

! 


1 7G° 


1 77° 


1 >-Q° 
1 ' 


79° 


1 80° 


1 81° 1 82° 


' 





1970 2957 


974 3701 


978 1476 


j981 6272 


9343 073 9376 8S3 9902 631 


60 


1 


3661 


4355 


2080 


1 6826 


582 9877 838 9903 085 


59 


2 


4363 


5008 


2684 


7380 


9849 C86 ' 792 j 489 


58 


8 


5065 


I 5660 


3287 


7933 


j 589 9378 245 1 891 


57 


4 


5766 


6311 


8889 


8485 


19350 C91 1 697 9904 293 


56 


5 


6466 


6962 


449 J 


9037 


1 593 9879 148 694 


55 


6 


1 7165 


7612 


5o9d 


i 9587 


9851 093 599 9905 095 


54 


7 


7863 


8261 


56S9 


982 0137 


1 593 


9330 048 i 494 


53 


8 


8561 


89.9 


6288 


1 0686 


9352 092 


! 497 1 893 


52 


9 


9258 


9556 


6886 


i 1284 


590 


i 945 9906 290 


i 51 


10 


9953 


975 0238 


74S3 


! 1781 


9853 087 


,9881 892 637 


50 


11 


971 0649 


0849 


8*79 


i 2327 


583 


1 833 9907 083 


49 


12 


134:3 


1494 


81374 


1 2873 


9354 079 


9882 234 | 478 


48 


13 


2036 


2183 


9268 


8417 


574 


! 723 1 873 


47 


14 


2729 


2781 


9362 


! 8961 


.9855 068 


,9838 172 9908 266 


46 


15 


8421 


3423 


^79 0455 


4504 


1 561 


1 615 1 659 


45 


16 


4112 


4065 


1047 


5046 


19856 053 


9884 057 9909 051 


44 


17 


48:32 


4706 


1638 


5587 


i 544 


j 498 ! 442 


43 


18 


5491 


5345 


2228 


6123 


9857 C35 


1 939 ! 832 


42 


19 


6180 


59S5 


2818 


6668 


524 


,9335 378 9910 221 


41 


20 


6867 


6623 


3406 


7206 


9358 013 


1 817 i 610 


40 


21 


7554 


7260 


8994 


7744 


501 


9886 255 , 997 


89 


22 


8240 


7897 


4581 


8282 


933 


692 ;9911 384 


88 


23 


S926 


8533 


5167 


8818 


9359 475 


9887 128 1 770 


87 


24 


9610 


9168 


5752 


9353 


960 


564 :9912 155 


36 


25 


972 0294 


9802 


6337 


9833 


9860 445 


993 


540 


35 


26 


0976 


976 0435 


6921 


933 0422 


929 


9888 432 


923 


34 


27 


1658 


1068 


7504 


C955 


9861 412 


865 


9918 306 


33 


28 


2339 


1699 


8C83 


1487 


894 


9389 297 


688 


32 


29 


3020 


2330 


8668 


2019 


9862 875 


728 


9914 069 


31 


30 


8699 


2930 


9247 


2549 


856 


9890 159 


449 


30 


31 


48T8 


3589 


9827 


3079 


9863 336 


583 


828 


29 


32 


5056 


4218 


930 0405 


8608 


815 


9391 017 


9915 206 


28 


33 


5733 


4845 


0983 


4136 


9S34 293 


445 


584 


27 


34 


6409 


5472 


1560 


4663 


770 


872 


961 


26 


35 


7U84 


6093 


2136 


5189 


9365 246 


9S92 298 9916 337 


25 


8Q 


7759 


6723 


2712 


5715 


722 


723 ! 712 


24 


37 


8432 


7347 


82S6 


6239 


9360 196 


9S93 148 ,9917 C86 


23 


38 


9105 


7970 


8860 


6763 


670 


572 459 


92 


39 


9777 


8593 1 


4433 


■ 7236 


9867 143 


994 832 


21 


40 


973 0449 1 


9215 


5005 


7808 


615 


9394 416 9918 204 


20 


41 


1119 


9886 


5576 


8330 


9368 087 


838 574 


19 


42 


1789 


977 0456 


6147 


8850 


557 


9395 258 944 


18 


43 


2458 


1075 


6716 


9370 


9369 027 


677 9919 314 \ 


17 


44 


3125 


1693 


7285 


9880 


496 


9396 096 1 632 ! 


16 


45 


3793 


2311 


7853 


984 0407 


964 


514 9920 049 i 


15 


46 


4459 


2928 


8420 


0924 


9370 431 


931 416 


14 


47 


5124 


3544 


8936 


1441 


897 


9897 347 782 


13 


48 


5789 


4159 


9552 


1956 


9371 363 


762 9921 147 


12 


49 


6453 


4773 


931 0116 


2471 


827 


9898 177 611 


11 


50 


7116 


5387 


0680 


2935 


9372 291 


690 874 


10 


51 


7778 


5999 


1243 


3493 


754 


9899 008 9922 237 


9 


52 


8439 


6611 


1805 


4010 


9373 216 


415 


599 


8 


53 


9100 


7222 


2366 


4521 


678 


826" 


959 


7 


54 


9760 


7832 


2927 


5032 


9874 133 


9900 237 


9923 319 


6 


55 


974 0419 


8442 


3486 


5542 


598 1 


646 


679 


5 


56 


1077 


9050 


4045 


6050 


9875 057 


9901 055 


9924 037 


4 1 


57 


1734 


9658 


4603 


6558 


514 


462 


394 


3 


58 


2390 


978 0265 


5160 


7066 


972 


869 


751 


2 


59 


3046 


0871 


5716 


7572 


9876 428 


9902 275 


9925 107 


1 


60 


3701 


1476 


6272 


8078 


833 


681 


462 







18^ 


12° 


11° 


10° 


9° 


8° 


1° 


' 








Nat 


UEAL Cosines. [ 



NATtJRAL TaXGEKTS. 



470 



{ ' 76° 77^ 


78° 


79° 


80° 


81° 


82° 


' 


'4-01 07809'4-33 14759|4-70 40301 


5-14 45540!5-6 712818'6-8 18751; 


) 7-1 153697 60 


1 575701 7231 ( 


) 4-71 18686 


5-15 25557 


80944( 


256601 


304193 59 


2 4-02 07446'4'34 3001^ 


I 81256 


'5-16 05813 


906894 


8761261 455808! 58 


3 57440 8786( 


) 4-72 49012 


86811 


5-7 00866S 


496091 


60705C 


57 


4 4-03 07550:4*35 4586] 


4'73 1695415-17 67051 


101250 


616502 


759437 


56 


5 577794-36m^0? 


85083 15-18 48035 


199173 


78735S 


912456 


55 


6 '4-04 081251 6229? 


4'74 58401 


5-19 29264 


297416 


858665 


7-2 066116 


54 


7 1 58590;4-37 20731 


|4'75 21907 


5-20 10788 


S95988 


980422 


220422 


53 


8. '4-05 091741 79317 


90603 


92459 


494SS9 


6-4102633 


375878 


52 


9 


59877 4'38 38054'4-76 59490 


5-21 74428 


594122 


225301 


530987 


51 


10 


4-06 10700 9C940 


4-77 28568 5-22 56647 


693688 


848428 


687255 


50 


11 


01643 4-3955977 


978375-2339116 


793588 


472017 


844184 


49 


12 


4-07 12707 4-40 15164 


4-78 67300 5-24 21836 


898825 


596070 


7-3 001780 


48 


13 


63892 74504 


4-79 86957 5-25 04809 


994400 


720591 


160047 


47 


14 


4-08 15199 4-41 33996 


4-80 06808 88035 


5-8 095315 


845581 


818989 


46 


15 


66027 93641 


76854 5-26 71517 


196572 


971043 


478610 


45 


16 
17 


4-09 18178!4-42 53439 
69852:4-43 13392 


4-Sl 47096 5-27 55255 
4-3217586 5-28 89251 


293172 
400117 


6'5 096981 
223396 


638916 
7999.'9 


44 
43 


18 


4-10 21 649 73500 


88174 


5-29 28505 


502410 


350293 


961595 


42 


19 


73569 4-44 33762 


4-83 59010 


5-80 08018 


605051 


477672 


7-4 123978 


41 


20 


4-11 25614 94181 


4-8430045 


92793 


708042 


605538 


287064 


40 


21 


7778414-45 54756 


4-85 01282 5-8177830 


811336 


788392 


45'J855 


89 


22 


4-12 30079 4-46 15480 


72719 5-82 68131 


915084 


862789 


615857 


88 


23 


82499 76379 


4-86 44359 5-33 48896 


5-9 019138 


992C80 


780576 


87 


24 


4-13 35046'4-47 87428 


4-87 16201 5-34 34527 


128550 


6-6121919 


946514 


36 


25 


877191 98636 


88248 5-35 20626 


228322 


252253 


7-5113178 


35 


26 


4-14 40519'4-48 60004 


4-88 60499 5-86 06993 


883455 


338100 


280571 


34 


27 


93446i4-49 21532 


4-89 82956 98680 


438952 


514449 


448699 


33 


28 


4-15 46501 


83221 


4-90 05620 5-37 8f)538 


544815 


646807 


617567 


32 


29 


09685 


4-50 45072 


784915-38 67718 


651045 


778677 


787179 


31 


30 


4-1652998 


4-51 07085 


4-91 51570'5-89 55172 


757644 


911562 


957541 


30 


31 


4-17 06440 


69261 


4-92 24S59'5-40 42901 


864614 


6-7 044966 


7-6 128657 


29 


32 


60011 


4-52 81601 


9S35S;5-41 80906 


971957 


178891 


800538 


28 


33 


4-18 18713 


94105 


4-93 72068'5-42 19188 


6-0 079676 


818341 


478174 


27 


34 


67546 


4-5356778 


4-94 45990|5-43 07750 


187772 


448318 


6465S4 


26 


35 


4-19 21510 


4-5419608 


4-95 20125 96592 


296247 


533826 


820769 


25 


36 


75606 


82608 


94474 '5-44 85715 


405103 


719867 


995785 


24 


87 


4-23 29835 


4-5545776 


4-96 6903715-45 75121 


514848 


856446 


7-7171436 


23 


38 


8419614-5609111 


4-97 43817 5-46 64812 


623967 


993565 


348023 


22 


39 


4-21 3S690 72615 


4-9818813 5-47 54788 


733979 


6-8131227 


525366 


21 


40 


93318'4-57 36287 


94027 5-48 45052 


844881 


269437 


708506 


20 


41 


4-2248080 4-5800129 


4-99 69459 5-49 85604 


955174 


403196 


882453 


19 


42 


4-23 02977 64141 


5-00 451115-50 26446 


6-1 066360 


547503 


7-8 062212 


18 


43 


5801 9 4-59 28825 


5-01 2C9S4!5-51 17579 


177948 


687373 


242790 


17 


44 


4-24 13177 


92680 


97078 5-52 G9005 


280923 


827807 


424191 


16 


45 


68482 


4-60 57207 


5-02 78895 5-58 00724 


402303 


968799 


606423 


15 


46 


4-25 23923 


4-61 21908 


5-08 49935 


92740 


5151.85 


6-9 110359 


789489 


14 


47 


79501 


86788 


5-04 26700 , 


5-54 85052 


628272 


252439 


973396 


13 


48 


4-26 3521 S 


4-62 51882 


5-05 08690. 


5-55 77663 


741865 


895192 


7-9 15S151 


12 


49 


91072 4-63 17056 


80907 . 


5-56 70574 


855867 


588478 


848758 


11 


50 


4-27 47066 82457 


5-06 68852 5-57 687861 


970279 


682335 


580224 


10 


51 


4-28 C3199I4-64 48084 


5-07 36f!2515--58 57302 


5-2(85106 


826731 


717555 


9 


52 


59472 4-65 13788! 


503 1392S 5-59 51121 


200847 


971306 


905756 


s 1 


53 


4-29 15885 


79721 


92061 5-60 45247 


816007 7-0 11 74 n' 


S\) 094835 


7 1 


54 


72440 


4-6645832 


5-09 7042615-61 39680 


482086 


2636621 


284796 


6 - 


55 


4-30 29186 


4-6712124' 


5-10 4 9. 124 '5-62 8M21 


648-)88 


410482 


475647 


^ 1 


56 


85974 


78595' 


5-11 27S55'5-68 294741 


665515 


557905 


667894 


^ 


57 


4-31 42955 4-6S 45248 5-12 06921 15-64 24S3S 


782863 


705934 


860042 


3 i 


58 


4-32 00079 4-09 120831 86224 '5-65 20516 


900651 


85457-'5 8-1 (>53599! 


'i 


59 


57347 79100 5 • 1 8 65763 5 -66 1 6509 


i'3 018866 7-1 008326 


243071 


1 


60 


4-33 14759 4-70 46301 5-14 45540 5-67 12818 


187515 158697 


44^3464 





' 


ir \ ir 1 ir 1 lo" 


9" 8° 


7^ ' 1 


Natural Cotangents. 



480 



Natural Sinks. 



' 


88° 


84° 85° 


86° 


87° 


88° 


89° 1 ' 





9925 462 


9945 219 9961 647 |9975 641 


9986 295 


9993 908 


9998 477 


60 


1 


816 


523 


9962 200 


843 


447 


9994 009 


527 


59 


2 


9926 169 


825 


452 


997G 045 


598 


110 


577 


58 


3 


621 


9946 127 


704 


245 


748 


209 


625 


57 


4 


873 


428 


954 


445 


898 


308 


673 


56 


5 


9927 224 


729 


9903 204 


045 


9987 046 


405 


720 


55 


« 


573 


9947 028 


453 


843 


194 


502 


76^ 


54 


7 


922 


827 


701 


9977 040 


840 


598 


812 


53 


8 


9928 271 


625 


948 


237 


436 


693 


856 


52 


9 


618 


921 


9904 195 


433 


681 


788 


900 


51 


10 


965 


9948 217 


440 


627 


775 


881 


942 


50 


11 


9929 310 


513 


685 


821 


919 


974 


984 


49 


12 


655 


807 


929 


9978 015 


9933 061 


9995 066 


9999 025 


48 


1.3 


999 


9940 101 


9065 172 


207 


203 


157 


065 


47 


14 


9930 342 


393 


414 


899 


844 


247 


105 


46 


15 


685 


635 


655 


589 


484 


336 


143 


45 


16 


9931 026 


976 


895 


779 


023 


424 


181 


44 ! 


17 


367 


9950 266 


9966 135 


963 


761 


512 


218 


43 


18 


706 


556 


374 


9979 156 


899 


599 


254 


42 


19 


6932 045 


844 


612 


843 


9989 085 


G84 


289 


41 


20 


884 


9951 182 


849 


530 


171 


770 


823 


40 


21 


721 


419 


9967 095 


716 


306 


854 


857 


89 J 


22 


9933 057 


705 


321 


900 


440 


937 


889 


88 


23 


893 


990 


555 


9980 084 


573 


9996 020 


421 


37 


24 


723 


9952 274 


739 


267 


706 


101 


452 


36 


25 


9934 062 


557 


9968 022 


450 


887 


182 


^82 


35 


26 


895 


840 


254 


631 


963 


262 


511 


34 


27 


727 


9958 122 


485 


811 


9990 193 


841 


589 


as 


28 


9935 058 


403 


715 


991 


227 


419 


567 


82 


29 


389 


633 


945 


9931 170 


855 


497 


593 


31 


30 


719 


962 


9969 173 


848 


482 


573 


619 


30 


31 


9936 047 


9954 240 


401 


525 


6v9 


6.9 


644 


29 


32 


375 


518 


623 


70l 


734 


724 


668 


28 


83 


703 


795 


854 


877 


859 


7L'3 


602 


27 


34 


9937 029 


9955 070 


9970 080 


9982 052 


983 


8a 


714 


26 


35 


855 


845 


804 


225 


9391 106 


943 


736 


25 


38 


679 


620 


528 


893 


228 


9097 015 


756 


24 


37 


9938 003 


893 


750 


570 


850 


€80 


776 


23 


38 


826 


9956 165 


972 


742 


470 


156 


795 


22 


39 


648 


437 


9971 193 


• 912 


590 


224 


813 


21 


40 


960 


7(i8 


413 


9983 082 


709 


292 


881 


20 i 


41 


9939 290 


973 


633 


250 


827 


360 


847 


19 ' 


42 


610 


9957 247 


851 


418 


944 


426 


863 


13 


43 


928 


515 


9972 069 


485 


9992 060 


492 


878 


17 


44 


9940 246 


783 


236 


751 


176 


556 


892 


16 


45 


563 


9958 049 


502 


917 


290 


620 


905 


15 j 


46 


830 


315 


717 


9984 081 


404 


633 


917 


14 1 


47 


9941 195 


580 


931 


245 


517 


745 


928 


13 i 


48 


510 


844 


9973 145 


4o8 


629 


807 


939 


12. i 


49 


823 


9959 1G7 


857 


570 


740 


867 


949 


11 1 


50 


9942 136 


370 


569 


781 


851 


927 


958 


10 i 


51 


448 


031 


780 


891 


960 


986 


9QQ 


9 


52 


760 


892 


990 


9935 050 


9993 069 


9998 044 


973 


8 


53 


9943 070 


9980 152 


9974 199 


209 


177 


101 


979 


7 


54 


379 


411 


408 


367 


234 


157 


985 


6 


55 


688 


669 


615 


524 


890 


213 


989 


5 


56 


998 


926 


822 


630 


495 


267 


993 


4 


57 


9944 3j3 


9961 183 


9975 028 


835 


600 


321 


996 


3 


58 


609 


438 


233 


989 


704 


374 


993 


2 


59 


914 


693 


487 


9986 143 


806 


426 


t-0000 000 


1 


60 


9945 219 


947 


641 


295 


908 


.477 


000 





' 


6° 


5° 


4° 


8° 


2° 


1° 


0° ' 1 


Natubal Cosines. I 



!N^ATURAL Tangents. 



481 



' 1 83° 


84° 


85° 


86° sr 88° 


89° 


' 


8-1 443464'9-5 143645' 11-430052' 14-300666' 19-08113t' 28-63625,^ 


57-289962 60 


1 68973GI 410613 4684T4 360696,' 187980 87TC8S 


58-261174 59 


2 83794 


679068 507154 421230 295922 29-122002 


59-265872 58 


3 8-2 035239' 949022 546(i93 4822731 405133 371l0f 


60-305820 57 


4 


2343819-6 220486; 585294 5438331 515584' 624499 


61-882905 56 


5 


43448.- 


49347L 


' 624761 605916 627296' 882298 


1 62-499154 55 


6 


635547 


768000 


; 664495 668529 74(;291 30-14461fl 


i 63-656741 54 


T 


837579 9-7 044075 


7045001 731679, 854591] 41158C 


64-858008 53 


8 


8-30405861 321713 


744779 795372 970219 683807 


66-105473 52 


9 


244577 


600927 


785333,' 859616; 20-087199 959928 


67-401854 51 


10 


4495581 881732 


826167 


924417 


205553 31-241577 


68-750087 50 


11 


655536 9-8 164140 


867282 


989784 


825308 528892 


70-153346 49 


12 


862519 


1 448166 


9jS6S2 15-05572S 


44-6486 820516 


71-615070 48 


13 


8-4 070515 


733823 


950370 


12-2242 


569115 32-118G99 


78-188991 47 


14 


2795319-9 021125 


' 992349 189349; 693220; 42129c 


74-729165 46 


15 


489578 


310088 12-034622 


, 257052 81S828 730264 


76-890009 45 


16 


700651 


600724 


077192 


325358 945966 83-045173 


78-126342 44 


17 


912772 


89305D 


120062 


894276121-074664" 366194 


79-943430 48 


18 18-5 125943 


10-018708 


163236 


; 463814 


204949 6935C9 


81-847041 42 


19 


840172 


048-383 


206716; 533931 


83685t- 34-027308 


83-843507 41 


20 


555468 


078031 


250505 


■ 604784 


4704011 867771 


85-939791 40 


21 


771888 


107954 


294609 


676233 


605680' 715115 


88-143572 39 


22 


989290 


18S054 


339023 


748337 


742569 85-069546 


80-463386 38 


23 


8-6 2;)7S33 


168332 


8S3768 


821105 


8812511 431282 


82-908487 87 


24 


427475 


19S789 


423831 


894545 


22-021710' 800553 


85-489475 36 


25 


648223 


2294-28 


474221 


968667 


163980 36-177596 


88-217943 35 


26 


870088 


26)249 


519942; 16 043482 


808097, 562659 


101-10690 


1 84 


27 ;8-7 093077 


291255 


565997 


118998 


454096 956001 


104-17094 


1 33 


28 


817198 


822447 


612390 


195225 


602015 87-857892 


107-42648 


1 32 


29 


542461 


853827 


659125 


272174 


751392 768618 


110-89205 


i31 


80 


768874 


385397 


706205 


849855 


9::3766 88-188459 


114-58865 


' 30 


31 


996446 


417158 


753634 


428279 23-057677, 617738 


118-54018 


29 


82 


8-8 225186 


449112 


801417 


607456 


213666 89 056771 


122-77896 


28 


83 


455103 


481261 


849557 


587398 


8717771 505895 


127-82184 


27 


34 


686206 


513607 


893058 


668112 


532052 965460 


132-21851 


26 


85 


918505 


546151 


946924 


749614 


694587 40-485837 


187-50745 


25 


36 


8-9 152009 


578895 


993160 


831915 


8592771 917412 


148-23712 


24 


87 


386726 


611841 


13-045769 


915025 24-r.26320' 41-410588 


149-46502 


23 


88 


62266S 


644992 


095757 


998957 


195714 915790 


156-25908 


22 


89 


859843 


678348 


146127 


17-083724 


867509 42-433464 


163-70019 


21 


40 


9-0 098261 


711913 


196S83 


169337 


541758 964077 


171-88540 


20 


41 


337983 


745687 


248031 


2558G9 


718512 43-508122 


180-93-220 


19 


42 


578867 


779573 


299574 


843155 


897826 44-066113 


190-98419 


18 : 


43 


821074 


813872 


351518 


431385 25-079757 688596 


202-21875 


17 ' 


44 9-1064564 


848288 


408867 


520516 264361 45 226141 


214-85762 


16 1 


45 


309348 


882921 


456625 


610559 451700! 8-29351 


229-18166 


15 ! 


46 


555436 


917775 


509799 


7015-29 641832 46-448862 


245-55198 


14 ! 


47 


8 )2S3S 


952850 


568391 


793442 834S23 47-085343 


264-44080 


13 


48 9-2 051564 


988150 


617409 


886310 23-080736' 739501 


286-47773 


12 1 


49 


3016-27 


11-023676 


671856 


980150 


229638 48-412084 


812-52137 


11 i 


50 


553035 


059431 


726738 


18-074977 


481600 49-103881 


843-77371 


10 1 


51 


805802 


095416 


782060 


170807 


686690 815726 


881-97099 


9 j 


52 9-3 059936 


131635 


837827 


267654 


844984 50-548506 


429-71757 


8 ' 


53 


815450 


16S0S9 


894045 


865537 


27-056557151-308157 


491-10600 


T ■ 


54 


572355 


204780 


950719 


464471 


271486; 52-080673 


572-95721 


6 


55 


830663 


241712 


14-007856 


564478 


489S53 8821 09 


687-54887 


6 


56 9-4 090384 


278885 


065459 


665562 


711740| 53-708587 


859-43680 


4 i 


57 


351531 


816304 


123586 


767754 


937233 54-561300 


1145-9153 


3 


58 


614116 


853970 


182092 


871068 28-166422 55-441517; 


1718-8732 


2 


59 


878149 


891885 


241184 


975523 


899397 56-350590 


8437-74()7 


1 


CO 


9-5 143645 


430052 


800666 


19-081137 


636253 57-289962 


Infinite. 





' 


6° 


5° 


4° 


3" 


2° 1 r 


0° 


/ 






Natui 


lAL Cotangents. 





41 



i 





LOGAEITHMS 


OF NUMBEES 






FROM 1 TO 10,000. 






N. 


Log. 


N. 
26 


Log. 


|n. 


Log. 


N. 


Log. 


1 


0-000000 


1-414973 


51 


1-707570 


76 


1-880814 


2 


0-301030 


27 


1-431364 


52 


1-716003 


77 


1-886491 


3 


0-477121 


28 


1-447158 


53 


1-724276 


78 


1-892095 


4 


0-602060 


29 


1-462398 


54 


1-732394 


79 


1-807627 


5 


0-698970 


30 


1-477121 


55 


1-740363 


80 


1-903090 


6 


0-778151 


31 


1-491362 


56 


1-748188 


81 


1-908485 


7 


0-845098 


32 


1-505150 ' 


57 


1-755875 


1 82 


1-913814 


8 


0-903090 


33 


1-518514 ; 


58 


1-763428 


1 83 


1-919078 


9 


0-954243 


34 


1-531479 


59 


1-770852 


1 84 


1-924279 i 


10 


1-000000 


35 


1-544068 


60 


1-778151 


1 85 


1-929419 I 


11 


1-041393 


36 


1-556303 


61 


1-785330 


1 86 


1-934498 


12 


1-079181 


37 


1-568202 ! 


62 


1-792392 


! 87 


1-939519 i 


13 


] -113943 


38 


1-579784 i 


63 


1-799341 


1 88 


1-944483 i 


14 


1-14G128 


39 


1-591065 : 


64 


1-806180 


i 89 


1-949390 ! 


15 


1-176091 


40 


1-602060 1 


65 


1-812913 


1 90 


1-954243 


16 


1-204120 1 


41 


1-612784 ; 


66 


1-819544 


, 91 


1-959041 


17 


1 - 230449 


42 


1-623249 


67 


1-826075 


92 


1-963788 


18 


] 255273 


43 


1-633468 1 


68 


1-832500 


93 


1-968483 


19 


1-278754 


44 


1-643453 1 


69 


1-838840 


94 


1-973128 


20 


1-301030 i 


45 


1-653213 


70 


1-845098 


95 


1-977724 


21 


1-322219 ^ 


46 


1-662758 1 


71 


1-851258 


96 


1-982271 


22 


1-312423 


47 


1-672098 f 


72 


1-857332 


97 


1-986772 


23 


1-361728 


48 


1-681241 


73 


1-863323 


98 


1-991226 


24 


1-380211 


49 


1-090196 


74 


1-869232 


99 


1-995635 


25 j 


1-397940 


50 


1-698970 


75 


1-875061 


100 J 


2-000000 



484 



Logarithms of Lumbers. 



i>ro. 



100 

1 

2 
3 
4 
5 



110 

1 
2 



000000000434 000868 
4321 1 4T51i 6181 
8600| 9026 9451 

012837 013259 013680 
T083! T451 

021189 021603 022016 
53061 5T15 6125 
9384! 9789 030195 
3424033826 4227 
7426j 7825; 8223 

C41893C41 787 042182 
5323 i 5714 6105 



120 

1 



9 

180 
1 
2 
3 
4 
5 
6 
7 



140 
1 
2 

8 
4 
5 
6 

T 



150 
1 
2 
8 
4 
5 



No. 



9 iDiff. 



013(29 003461 0{;8891 432 
73211 7748! 8174 428 

011570,011993 012415 424 
5779! 6197! 6616 420 
9947 020861 C20775 416 

024075 4486J 4896' 412 
81641 8571! 8978-4(8 

C8221 6X82619 083(21 1 404 
6230! 6(^29 7(28 400 

04C2(;7M0602 C4( £98 897 

042576 0429G9 048362 048755 044148 C4454o!c44982' 893 

6495 6^85' 7275 76641 8(58 8442' 8830^ £90 

218' 9606 9993|050S8) 050766 051158 051588X51924 (528(9 (52094 886 

05307S 05S4()8 053846 4230 4613i 4996 53781 5760 6142! 6524 883 

69051 7286 7666! 8046' 8426! 88(i5' 9185! 9568| 9942 06(820:879 

060698061075 C61452!061829 062206 062582 0629581 068883;C687( 9. 4088! 876 

4458! 4832 5206 558')| 5953! 6826 6099! 7071 7443^ 7815' 873 

81861 8557 8928: 9298 96(iS 070(.88 0704(7X70776 071145 (71514 370 



001301 001734002166 002598 
5609 6088! 6466! 689' 
9876 01 0300; 010724 011147 

014100, 45211 49401 5860 
8284 8700! 91 ] 6' 9532 

022428 022841 023252 023664 
6533 6942' 7350 i 7757 

030600 031004 031408 081812 
4628 5029' 5480| 5830 
8620 9017 9414 9811 



71882 072250 0726171072985 078352, 
55471 5912 6276 6640! 7004! 



8718, 
73G8' 



4085 

rsi 



4451 
8(94 



I I ! I 

07918l!079543 079904'080266 C80628 CSC9S7 (81847,081707 



120574 120903 
88521 41' 
7105 7429 



4816 5182 866 

8457! 8819, 363 

082067(82420; 860 

5647i 6004' 857 

9198! 9552 355 



082785'083144 083503 8861' 4219 4576 4984 5291 
6860 6716 7071 7426 7781! 8186^ &490| e8^5 
9905 090258,090611 090933 091315091667 092018 C9287o;( 92721 093071! 852 

193422! 8772! 4122 4471; 4S20| 51C9, 5518' 5866! 6215J 6562' 849 
6910 7257 7604 7951' 8298 8C44! 8990! 9885 9681 1(0(26' 846 

100371 100715 101059 101403 101747 102(91 1(2^34 1(2777 IC8II9! 84(i2 843 
8804 4146 44S7i' 4828' 5109, 5510! 5851 619i: 6r8ll 6871! 841 
7210 7549 78S8| 8227i 8565! 89C8I 9241 1 9579 9916,llC258i 838 

110590 110926 111263 111599 111934 112270 112605 112940 118275 86(9 835 

113943 114277 114611ill4944'll527S 115611 1159481102761166(8116940 833 

271 7603 7934 8265! 8595: 8926! 9256! 9586' 9915120245 830 

12128l!l21560l218S8 12221 6 122544 122871 128198! 8525 828 



4504! 

7753I 



48301 

^076^ 



5156i 
8099 



64811 

8722' 



6806 

90-^ 5! 



6131| 

9868! 



64561 6781 825 



9690 



18001 2! 823 

8219! 821 

6403! 818 

9564! S16 

9879 140194'140508 140822 1 41186 141450 141763,142076 142889 142702' 814 



130834 130655 130977,131293 181 ('.19 131989 182260 182580 182900 
85891 3858| 41771 4496; 4814| 5183| 64511 5769, 6(86 
67211 7037 73541 76711 7.987! 8303! 8618| 8984! 9249 



6818 



143015! 33271 3689. 3951J 4263, 4574| 4885J 6196j 6607 

146128 146488 146748 147058 147367 147676 147985 148294 14S608'l48911 
9219; 95271 9835 150142 150449 150756 151063,151870 151676 151982 

152288152594152900! 3205^ 8510! 8815| 4120 4424| 4728| 5082 
5336; 5640! 5943 6246' 6549! 6852' 7154i 74571 77691 8061 
88621 8664! 6965' 9266' 95071 9868 160168 16C469 160769 161068 



311 

3(9 

807 
305 



161368 161667:161967 1622661 62564 162S63J 8161 8460 3758 4055 
4353 4650 4947, 5244' 6541 1 6838 6184 6430 6726 7022 
73171 7613! 7908 82G3' 8497 8792* 9(86! 938O 9674 9968 

170262 170565 170848 171141 171484 171726 172( 19 172811 172603 172895 
8186^ 3478! 3769^ 4060! 4351| 4641! 4932J 62221 55121 5802 291 

176091 176881 176670 176959 177248177586 177825 178113 178401 178689 289 
359 1 SOI 26 180413 180(-99 180986 181272481558 287 



301 
299 
297 
295 
293 



8889 
6674 
9490 



8977^ 9264 9552 

1818441821291824151827001 2985 8270! 35551 
4691 4975; 5259 5542 5825' 6108, 6891 
7521! 7803! 8084 8366i 8647i 8928! 9209, 

190882 190612 190692 191171 191451 191780 192010;i922&9 192567 
3125 3403i 3681 8959! 42871 4514; 4792 5069, 5846 
6900l 6176 6453 6729! 7005 728l! 7556| 7832| 8107 
86571 8932 9206 948l! 9755 200029 200303 200677 200850 201124 

201397 201670 201943 202216 202488, 2761| 3033| 8305| 3577 8848 



4123 4407 
69561 9289 
9771 190061 

2846 







1 



6 



285 
283 
281 
279 
278 
276 
274 
272 

Diff. 



Logarithms of Numbers. 



460 



No. 





1 


2 3 1 4 


5 


6 


7 8 9 Diff. 


160 


20412;) 204391 204663 204984 2052C4 2C5475 2G5746 


206016 206286 206556! 271 


1 


6826: 70961 73651 7634 7904^ 8173 8441 


8710 8979 9247, 269 


2 


9515; 9783 21005l'210319 210586 210853 211121 


21138S 211654 211921! 267 


3 


212188 212454:' 272f 


2986 8252 3518 3783 


4049 4314^ 4579 266 


4 


4844: 51091 537S 


5638; 59^!2 6166 6430 


6694 69571 7221 i 264 


5 


7484' 77471 801C 


8273' 8536^ 8793 9060 


9323' 9585' 9846; 262 


6 


220108 22:1370 220631 


220892 221153 221414 221675 


221936 222196 222456 261 


7 


2716j 2976 


8236 


3496 3755, 4015 4274 


4583, 4792: 5051 ! 259 


8 


53{t9i 5568 


5826 


6084' 63421 6600 6858 


7115; 7372' 7630 ! 258 


9 


7887 8144 


8400 


86571 89131 9170! 9426 


9682' 9938 230193 


256 


170 


230449 239704'23n980'231215'231470'231724 231979'282284 232488'232742 


255 


1 


2996, 3250 


3504 


3757| 4011; 4264 4517 47701 5023 5276 


253 


2 


55Q8 5781 


6033 


6235' 65371 6789 7041 7292' 75441 7795 


252 


3 


8)46' 8297 


8548 


8799 9949 9299 9550 9800 240050 240300 


250 


4 


240549 240799 241048 241297 241546 241 795 242044; 242293: 2541 


2790 


249 


6 


3038 82861 3534 


3782 40301 4277 4525 


4772! 5019 


5266 


248 1 


6 


55131 5759 6006 


6252 6499 6745 6991 


7237! "482 


7728 


246 1 


7 


7973' 8219 8464 


8709 8954' 9198 9443 


9687! 9932250176 


245 i 


8 


250420 250664 250908251151 251395 251638 2518811252125 252368 2610 


243 i 


9 


2353j 3096 3333J 3580| 3S22; 4064 4306r 4548| 4790 5031 


242 1 


18D 


255273 255514 255755 255996 256237256477 256718;25695s'25719S 257439 


241 1 


1 


7679, 7918! 8158! 8398 8637: 8877 9116 9355: 9594| 9833 


239 ; 


2 


26007l'2603l0 260548 260787 261025 261263 2615011261739 261976,262214 


238 1 


3 


24511 2688 


2925 


3162 3:3991 3636 8878 41C9' 4346 4582 


237 ! 


4 


4818J 5054 


5290 


5525, 5761! 5996 0232 6467i 67C2 C987 


235 ! 


5 


7172j 7406 


7641 


7875' 81101 8844' 8578 8812' 9046: 9279 


234 i 


6 


9513 9746 


9930 270213 270446270679 2719121271144 271377 271609 


233 ■ 


7 


271842 272074 


272306 


2538 2770 


8001 32331 3464! 3696! 3927 


232 


8 


4158 4889 


4620 


4850 5081 


5311 5542! 5772! 6002, 6232 


230 


9 


6462 6692 


6921 


7151 7380 


7609 7838| 8j67| 8296 8525 


229 i 


190 


278754 278982 


279211 279439'279667'279S95'280123 230351 '280578'2808O6 


228 ' 


1 


281033 2S1281 


281488 281715 231942 282169, 2396, 2622 


2849 3075 


227 1 


2 


3301 


3527 


3753 


3979 4205| 4431 4656; 4882 


5107 5332 


226 i 


3 


5557 


5782 


6007 


6232 6456 6681 6905; 7130 


7354 7578 


225 ; 


4 


7802 


8026 


8249 


8473 8696 8920 9143 9866 


9589 9812 


223 ! 


5 


290035 290257 


290480 290702 291-925 291147 291369 291591 291813 292034 


222 1 


6 


2256 


2478 


2699 


2920 


3141 33631 3584 8804! 4025! 4246 


221 ' 


7 


4466 


4687 


4907 


5127 


5347 


5567 5787| 6007 6226! 6446 


220 ; 


8 


6665 


6884 


7104 


7323 


7542 


7761 7979 8198 8416; 8635 


219 1 


9 


8853 


9071 


9289 9507 


9725 


9943 80C161 300378 300595 300813 


218 j 


200 


301030 


301247 


301464 301681 


301898 302114'302831 302547 3&2764'3029SO 


217 


1 


3196 


3412 


3628 


3844 


4059. 4275, 4491 4706 


4921 5136 


216 


2 


5351 


5566 


5781 


5996 


6211 0425! 6639 6854 


7068 7282 


215 


3 


7496 


7710 


7924 


8137 


83511 8564' 8778 8991 


9204 9417 


213 


4 


9630 


9843 310056'310268 310481 810693 810006 811118 811330 311542 


212 


5 


311754 311966 


2177 


2389 2600 2812, 3023 3234 


8445! 8656 


211 


6 


3867 


4078 


4289 


4499 4710 4920 ol30! 5340 


5551! 5760 


210 


7 


5970 


618:') 


6390 


6599 6809 7018 72271 7436 


76461 7854 


209 


8 


8063 


8272 


8481 


8689 8898' 9106! 9314' 9522 


9730 9938 


208 


9 


320146 320354 320562 320769 320977 321184 321391821598 821805 822012 


207 


210 


322219 322426,322633'322889'323046 323252823458 323665'323871 824077 


206 


1 


4282 


4488 


4694 


4899 51(^5! 5310| 5516 5721 5926 6131 


205 


2 


6336 


6541 


6745 


6950 7155' 7359 7563 7767 7972! 8176 


204 


3 


8380 


8583 


8787 


8991 9194' 9398 9601' 9805 830008 830211 


203 


4 


330414 33061 7,330819 831022 331225 331427 831630 8318321 2034! 2236,' 


202 


5 


2438 


2640 


2842 


3044 


3246 34471 3649 3S50| 4051 1 


4253 


202 


6 


4454 


4655 


4856 


5057 


5257 


5458; 6658 5859 G059 


6260 


201 


7 


6460 


6660 


6860 


7060 


7260 


7459, 7659 7858| 8058 


8257 


200 


8 


8456 


8656 


8855 


9054 


9253 


9451 1 9650 9849,340047 n40246| 


199 


9 


340444 


340642 340841 


341039^341237 341435 341632 341830 2028 


2225 


198 

'mi 


No." 





l" ~ 


2 


" 3.1 ^4 1 6" 1 ^6 T 7 8 



41* 





486 






Logarithms 


OF Numbers. 






1 




No. 





1 2 


3 


4 5 


6 


1 


8 1 9 Diff. 


220 


342423 342620 342817 343014 343212 843409 343606 343802 343999 344196 197 




1 


4392 4589 4785, 4981, 5178 5374 5570| 5766| 5962 6157 196 




2 


6353 6549 6744' 6939 7135 7330 i 7525 7720 7915 8110 195 




8 


8305 8500 8694' 8889 9^83 927S' 9472' 96661 9860 350054 194 




4 


35024S 350442 350636 350829 351023 851216 851410 351603351 796 1989 193 




5 


2183 2375 2568, 2761 2954 3147{ 33^39 3532 


3724: 3916 193 




6 


4108^ 4301 4493 4685' 4876 5068 5260 5452 


56431 5834 192 




7 


6026 6217 6408: 6599 6790 6981 7172 7368 


7554! 7744 191 




8 


7935' 8125 8316^ 8506 8C96 8886 9076 9266 


9456' 9646 190 




9 


9S85 860025 360215 360404 360593 360783 360972 361161 '361350 361589 189 




230 


36172s'361917 362105'362294362482 362671 862859'36804s'363236 363424 188 




1 


8612: 3800 3988 4176 4363 4551 


4789; 4926 


5118! 5301 188 




2 


5488 5675 58621 6049 6236 6423 


66I0I 6796 


6983! 7169 187 




8 


7356' 7542 7729^ 7915 8101 8287 


84731 8659 


8845 9030 186 




4 


9216! 9401' 9587i 9772 9958 370143 


370328 370513:370698 370883 185 ! 




5 3T1068 8T1253 871437 3T1 622 3T1 806 1991 


2175' 2360 


2544, 2728 184 i 




6 


2912 8096 8280! 8464 3647; 3831 


4015 4198 


4382: 4565 184 




T 


4748 4932 5115, 5298 5481 


5664 


5846 6029 


6212 6394 183 




8 


6577: 6759 6942' 7124' 7306 


7488 


7670 7852 


80341 8216 182 




9 


8898! 8580 8761; 8943! 9124 


9306 


9487 9668 


9849 880030 181 




240 38021l'380892'3S0573'380754'380934'3Sllln 


381296 3S1476'381 656 8S1S37 181 ! 




1 


20171 2197 2377 2557, 2737 2917 


3C97 


3277 3456 3686 180 i 




2 


8815' 8995' 4174 4853' 4533' 4712 


4891 


5070 5249: 5428 179 




8 


5606; 5785; 5964 6142 6821 64991 6677 


6856 7084! 7212 178 




4 


7390! 7568 7746 7923 8101' 8279' 8456' 86841 8811 8989 178 i 




5 


91661 9343 9520' 9698 9875 390051890228 890405,390582,890759 177 j 




6 


390935 391112 391288 391464 8916411 1817 


1993 


2169 23451 2521, 176 j 




7 


2697 2873 3048 8224 8400' 3575 


3751 


3926 


4101 42771 176 | 




8 


4452' 4627! 4802, 4977i 5152 5326 


5501 


5676 


5850 6025 


175 




9 


6199 6874' 6548^ 6722 6896 7071 


7245 


7419 


7592 7766 


174 




250 


397940'398114'39S287 898461 898634 898808 898981 '3991 54 


399328i399501 


173 




1 


9674 9847 400020 400192 400365 400538 400711 400883 '401056 1401 228 


173 




2 


401401401578' 1745, 1917. 2089 226li 2483 2605! 2777; 29491 172 j 




3 


31211 8292 3464' 3685' 8807 8978! 4149 4820t 4492| 4663i 171 | 




4 


4834 


5005 5176 5346, 5517i 5688 5858! 6029 6199 


63701 171 i 




6 


6540 


6710 6881 1 7051 ! 7221' 7891 I 75611 7731 7901 


8070! 170 




6 


8240 


8410 8579 8749 8918 9087! 9257 9426 95951 9764! 169 ! 




7 


9933:410102 410271 410440 410609 410777 410946 411114 411288 411451' 169 




8 


411620 1788 1956! 2124' 2293 2461 2629 2796! 2964i 3132, 168 




9 


3800 8467 8635' 3803! 8970 4137. 4305 4472! 46391 4806 167 




260 


414973 415140 415307'415474'415641 '415808 415974 416141 4163081416474 167 




1 


6641 6807 6973 7189 7806 7472 7638 7804 7970; 8135i 166 




2 


8301 8467! 8633 8798 8964 9129 9295 9460 96251 97911 165 




3 


9956 420121 420286 420451 420616 420781 420945 421110 421275 421489: 165 




4 


421604 


1768! 1933 2097! 2261! 2426' 2590! 2754 


2918! 3082' 164 




5 


3246 


8410' 3574' 3737 3901 1 4065' 4228 4392 


4555 4718! 164 




6 


4882 


50451 5208 5371 5534-; 5697! 5860| 6023 


6186 6849 163 




7 


6511 


6674- 6836 6999! 7161 ' 7824' 7486 7648 


7811; 7973; 162 




8 


8135 


82971 8459 8621' 8783' 8944' 9106 9268! 9429' 9591| 162 




9 


9752 


9914^430075 430236 430398 430559 430720 430881 '431042; 431 203; 161 




270 431364'431525'431685'431846'432007 432167 432328'432488'432649'432809! 161 | 




1 


2969, 3180; 3290, 8450; 3610, 3770 3930| 4090 4249i 4409, 160 | 




2 


4569 4729 48Ss! 5048 5207 5367 5526: 5685 


5844; 6004' 159 




3 


6163 63221 6481 6640 6799 6957 7116! 7275 


7433! 7592! 159 




4 


7751 79091 80671 8226 8384^ 8542' 870l! 8859 


9017! 9175! 158 




5 


9333! 9491! 9648' 9S06 9964 440122 440279 440437 440594:440752: 158 




6 ' 


44091.9 441066 441224 4413S1 441538 1695 1852; 2009 21661 2328 157 




7 


2430 2637! 2793. 2950: 3106 3263 3419 8576! 3732; 8889 157 




8 


4045 


4201 4357! 4513 4669 4825 498l! 5137 5293! 5449 156 ! 




9 


5604 


5760 5915 60711 6226; 6882 6537] 6692 


6848 7003! 156 1 




No. 





1 2l3i4i5i6i7 


8 


9 iDift; 1 



Logarithms of Numbers. 



487 



No. 







1 



3 



iDiif. 



280 
1 
2 
8 
4 
5 
6 
7 
8. 
9 

290 
1 
2 
8 
4 
5 



300 
1 
2 



44T158 447318:447468 

8706 8861 1 9015 

450249 450403 450657 



1786, 1940 

8318 8471 

4845' 4997 

6366 6518 

7882, 8033 

9392' 9543 



2093 
3624 
5150 
6670 

8184 
9694 



310 
1 
2 
3 
4 
6 



491362 491502 491642 491782 491922 



460898 461048 461198! 

i62398'462548 462697 
3893, 4042,1 4191 
5383 6532! 5680 
6868 70161 7164 
8347 8495| 8643 
9322' 9969 470116 

471292 4714381 1585 
2756i 2903 3049 
4216 4362! 4508 
6671 5816J 5962 

1477121477266 477411 

8566i 87111 8855 
480007,480151480294 

1443 

2874 

4300 

6721 

7138 

8551 

9958 



447623 447778 
9170| 9324 

450711 450865 
2247| 2400 
3777i 893(1 
5302 6454 
68211 6973 
8836' 8! 87 
9845' 99£5 

461348 4614^9 



447933 448088 
94781 9633 

451018451172 
2653| 2706 
4082 4235 
5606 5758 
7125 7276 
8688! 8789 

460146460296 
1649| 1799 



448242 448397,448552 155 



9787| 9941 
451326 451479 
2859) 8012 
4387 4540 
5910 6062 
7428; 7579 
89401 9G91 
460447 460597 
1948 2C98 



450095 154 

1633 154 

3165 158 

4692 153 

6214 152 

7731 152 

9242 151 

460748 151 

2248 150 



1586 


1729 


1872 


2016 


8016 


3159 


3302 i 8445 


4442 


4585 


4727 


4869 


6863 


6005 


6147 


6289 


7280 


7421 


7563 


7704 


8692 


8833 


8974 


91141 


490099 


490239 


490380 


490520 



2900 

4294 

5683 

7068 

8448 

9824 
501059 501196 501383 
24271 2564 2700 
8791 8927 4063 



2760 
4155 
5544 
6930 
8311 
9687 



8040 
4483 

6822 
7206 
8586 



505150 505286 
65051 6640 
7856 7991 
9203' 9387 



505421 
6776 
8126 
9471 



462847 462997 463146 4G3296 463445 
4340 4490 4689! 4788 4936, 
6829 5977 6126| 6274 6428, 
7312j 7460 7608' 7750 7904! 
87901 8938 9r85! 9283 98801 
70263 470410 470557'470704 470851 
1782 1878 2025! 2171 2318 
3195: 3841 84871 3683 3779, 
4653| 4799 49441 5090 5285! 
6107 j 6252 63971 6542" 6687 

477555 477700 477844'4779S9 478188 478278478422' 
8999 9143 9287 9431 9576' 9719 9863 

480438 480582:480725 ^80869 481012 481156 481299 
2159! 2802 2445 2588! 2781 
8587i 3730i 8872 4015 4157 
6011! 6158 5296 5487 5579 
6430l 6672 6714 6855; 6997 
7845! 7986! S127 8269' 8410 
9255: 9896! 9537 9677! 9818 
490661 490801 49C941 491081 491222 

4920624922011492841 492481 '492621! 
3168! 85971 3787j S876| 4015 
4850| 4989! 51281 5267 5406 
6288 6876! 65l6i 6653 6791 
76211 7759! 78971 8C85 8178 1 
8999' 91871 9276! 9412! 9550 

60C374'5(;9511i5C0648 600785 50C922' 
17441 1880 2017 2154| 2291' 
31C9| 3246 3882 8518| 8655 
4471 4607 4748 4878! 6014; 



8179 
4572 
6960 
7344 
8724 
9962 500099 
1470 
2887 
4199 



610545 510679 510818 



1883: 2017 

3218' 8351 

4548 4681 

5874^ 6006 

7196^ 7828 



No. 



2161 
3484 
4813 
6189 

7460 

618514 518646 518777 
9828 9959 520090 

521138 521269 1400 
2444 2575! 2706 
3746, 3876! 4006 
6045 5174 5304 
6389 6469 6598 
7630 7759: 7888 
8917 9045; 9174| 

530200 630828 530466 ! 

"o~l~T~l""'^l 



3319! 
47I1I 

6099! 
7483; 
8862' 
500286 
1607| 
29781 
4335j 

505693 
7046: 
8395' 
9740! 

511081 
2418 
8760 
5079 
6403 
7724 



463594463744 150 

5085 5234 149 

6571 6719 149 

8052 8200 148 

9527! 9675 148 

470998 471145 147 

2464 2610 146 

3925| 4071! 146 

5381 1 5526 146 

6882: 6976 145 

145 
144 
144 
143 
143 
142 
142 
141 
141 
140 

140 

189 
189 
189 



506557 

6911 

8260 

9606 
510947 

2284 

3617 

4946 

6271 

7592 

518909 519040 
520221620353 



605828 
7181 
8530 
9874 

511215! 
2551 1 
8883] 
5211! 
6585: 
7855 1 



505964 506099 

! 78161 7451 

' 8664i 8799 

5100(9 610148 

1849| 1482 

26841 2818! 

40161 4149! 

63441 54761 

6668' 68001 

79871 8119! 



1661 
2966 
4266 
6563 
685() 
8145 
9430 
530584!530712 



1530 
2835 
4186 
5434 
6727 
8016 



519171 
52(1484 
1792 
8(96 
4396 
6693 
6985 
8274! 
9559 
530840! 



519303' 
520615 
1922 
8226 
4526 
6822 
7114 
8402 
9687 
530968 



519484 

520745 

2058 

8856 

4656 

69511 

7243 

8581 

! 98151 

531096 



506284 5068701 

I 75861 7721: 

I 8984 9068 

510277 610411 

1616 1760 

2951' 8084 

4282 44151 

5609! 6741 1 

6932! 7064' 

! 8251! 8882 

519566 5196971 
520876 521007 
2183 2314 
3486 8616 
4785, 4915 
6081 1 6210 
7872| 7501 
8660 87.^8 
9943 530072 
5812231 1851! 



3 



187 
136 
186 

136 
185 
185 
134 
134 
183 
183 
188 
182 
182 

181 
181 
181 
180 
180 
129 
129 
129 
128 
_128 



488 



Logarithms of IS^umbers. 



No. 





1 


1 2 


3 


1 4 


1 5- 


6 


7 


8 1 9 


Diff. 

'\ 128 


8-13 J531479 5316jr 531784 581862 53199;) 582117 532245 582872 582500 58262- 


1 


2754: 28S2I 80j9 


318(] 


82641 8891! 3518 


8645 3772 8899 127 


2 


4)26 


4153 4230 


4407 


4584i 4661 i 4787 


4914 5041! 6167i 127 


8 


5291 


5421 5547 


5674 


5800 


5927 6053 


6180! 6306' 6432' 126 


4 


6553 


6635' 6311 


6937 


7068 


7189 7815 


7441 1 75671 7693 126 


5 


T319 


7945! 8371 


8197 


8822 


8448! 8574 


8699 8825 8951 126 


6 


9)7o 


9232 9327 


9152 


9578 


9703 93291 9954 540079 54(204 125 


7 


54 >.JJ9 540455 5405S3 540705 54^830 540£55 541080 541205 1880. 1454 125 


8 


157i) 1704! 1829 1958 2i)78 2208 2827 2452 2576 27a 125 


9 


2S25 295J 8374 8199 3328 8447| 3571 3696 38201 8944 124 


33) 


544)63 544192 544316 544440 544564 544688 544S12 544936 545060 545188' 124 


1 


53)7 


5131 


5555! 53 73 5832 


5925 60491 6172 6296 6419 124 


2 


6543 


6833 


6789, 6913! 7086 


7159' 7282| 7405 7529 7652 123 


8 


7775 


7393 


8321 8114' 8267 


8389 8512| 8635 8753 88^ 123 


4 


9))3 


9123 


9249! 9371! 9494 


9618 9789 9861 9984 551106 123 


5 


55 322355 )35l'55)478 550595 550717 550840 550962 551084551236 1823 122 


6 


1453 


1572 1391! 1313! 1938, 2060! 2181 


2803 2425 2547 122 


T 


21)63 


2793 2911 


3033 


81551 3276 3398 


8519 3640' 8762 121 


8 


8333 


4004 4123 


4247 


48681 4489! 4610 


4871 i 4852 4973 121 


9 


539i 


5215 5333 


5457 


6578| 5699! 5820 


59401 6061' 6182 121 


839 


5533)3'555423 553544'556684'556785'5569a5'557026'55T146 557267 557887; 120 


1 


75)7 


7627 7743: 7833 7938! 8108. 8228| 8849 8469 8589 120 


2 


8739 


8329 8918' 9368' 9183' 9808' 9428' 9548 9667 9787 120 


3 


93/7 


580323 533143 560235 563335 560504 56(j624 560748 560863 56C9S2 119 


4 


5311 ,1 


1221 134); 1459! 1573! 1698i 1817 


1936 2055 2174 119 


5 


2i33 


2412 253i 


2353 


2769| 2837 


3006 


8125! 3244 8862 119 


6 


8i81 


3630' 3713 


8337 


3955 4074 


4192 


4311 


4429 4543 119 


7 


4333 


4734 4933 


5321 


5189 5257 


5876 


5494 


5612; 5780 118 


8 


58 13 


5983 6)84' 6202 


6820| 6487 


6555 


6678 


6791 69G9 118 


9 


7023 


T144' 7232 7379 


74971 7614 


7782 


7849j 79671 8oS4 118 


370 


5S8232'568319 568436'568554'56867l'568788'568905'569028'569140 569257! IIT 


1 


9374 


9491' 9308! 97251 9812; 9959 570076 57ol93 5708(>9 570426' 117 


2 


570513 


570633 570773 570393 571010 571123 


1248 


1859, 1476 1592; 117 


3 


1709 


1825 1942 


2353 


2174 


2291 


2407 


2528 2689' 2755 


116 


4 


2872 


2933! 3104 


8220 


3886 


3452 


3568 


8684! 3800 ! 8915 


116 


5 


4331 


4147 4283 


4879 


4494 


4610 


4726 


4841 


4957 5072 


116 


6 


5133 


5303' 5419 


5584 


5650 


5765 


5880 


5996 


6111! 6226 


116 


r 


6341 


64571 6572 


6637 


6302 


6917 


7082 


7147 


7262 7377 


115 


8 


7492 


76071 7722 


7883 


7951 


8068 


8181 


8295 


8410| 8525 


115 


9 


8339 


8754 8363 


8983 


9097 


9212 


9326 


9441 


9555; 9669 


114 


383 


579784 


579398530012 580126'58024l'58085A580469'580588'580697 580811 


114 


1 


533925 581039 1153! 1267 


1381 1495 


1618, 1722! 1886 1950 


114 


2 


2363^ 21771 22911 2434 


2518 2681 


2745! 2858! 2972' 30S5 


114 


3 


8199 8312' 34231 3539 


3652 3765 


38791 8992 


4105 4218 


118 


4 


43311 4444' 4557J 4670 47S8| 4896 


5019 5122 


5235 5348 


118 


5 


5181' 5574' 5683 5799i 59121 6024 


6137 


6250 


6862 6475 


118 


6 


6587i 67001 6812 


6925! 7087 7149 


7262 


7874 


7486 7599 


112 


T 


7711! 7823 7985 


80471 8160 8272! 88841 8496 


8608' 8720 


112 


8 


8332 8944' 9356 


9167 92 r9: 9891! 95o8! 9615 


9726 9888 


112 


9 


9953 590061 593178 590234 590893 590507 590619 590730 590842 69u958 


112 


390 


591065 591176 591237 591899'591510 591621 591732 591848 591955 592066 


111 


1 


2177 2283 2399j 2510! 2621i 2782 2848i 29541.8064 3175 


111 


2 


8233 83971 3503 


8618 8729 3840: 8950| 


4061 


41711 4282 


111 


3 


4393 4503 4614 


4724 


4834! 49451 5055 


5165 


6276! 5388 110 


4 


5193 5636' 5717 


5827 


6987 6047 1 6157 


6267 


68771 6487 


110 


5 


65971 67071 6817 


6927 


7087 71461 7256 


7866 


7476! 7583 


110 


e 


7695, 7305 7914 


8024 


8134 8248! 8858 


8462 


8572 8681 


110 


T 


879l! 8900! 9009 9119 


9228' 98371 9448 


9556 


9665' 9774 


109 


8 


9383 9992 600101600210 600819 60042S 600587 600646600755 600864 


109 


9 


500973 6010821 1191| 1299, 1408, 1517i 1625 1734| 


1848, 1951 


1C9 


No. 


i) 1 


1 1 


'2. 


3 


4 1 


5 1 


6 1 


1'\ 


8 1 9 


Diif. 



Logarithms of Numbers. 



489 



No. 



Diff. 



400 
1 
2 
3 
4 
5 
6 
T 
8 
9 

410 
1 
2 
3 
4 
5 



420 
1 
2 
3 
4 
5 



430 
1 



602060 



602169 602277 602386 602494 6026C8 602711 602819 602928 603086| 



3253 



5413 



3361 
4442 
5521 
6596 



3469, 3577 


3686 


8794 


3902 


4550 4658 


4766 


48741 4982 


5628' 5736 


5844 


5951 


6059 


6704' 6811 


6919 


7026 


7133 


7777 78S4 


7991 


8098 


8205 


8847 8954 


9j61 


9167 


9274 


9914 610021 


610128 610234 610341 



3144 
4226 
5305 
6381 

7455' 7562 1 7669 
8526' 8633 8740 
9594' 9701 1 9808 
610660 610767 610873 610979, lOSOl 11921 12981 1405 



1723i 1829, 1936| 2042J 2148 

612784'612890 612996 613102 618207 
8842, 39471 4053; 4159, 4264 
48971 5003 5108 5213! 5319 
59501 6055 6160 
7000 7105 7210 
8048 ■ 8153 8257 
90931 9198! 9302 

620186 620240 620844 620448 620552 
11761 128l| 1384i 1488[ 1592 
2214! 2318: 2421 1 2525 2628 



6265 6370 

7815 7420 

8862 8466 

9406: 9511 



2254 2360 2466 
613818 018419 613525 



4ulOi 4118 

5089' 5197 

6166' 6274 

7241 7846 

8812 i 8419 

9381 ! 948S 
610447 61055^ 

1511i 1611 

2572; 267& 

613680 618738 



4370! 4475[ 4581 4686 47! 

5424 5529; 5634 5740| 5845 

6476 6581 6686 6790 6895 

7525 7629 7734 7889 7943 

8571 80761 8780 8884' 89S9 

9615! 97191 9824 9928 620082 

620656 G20760 620864 620968 1072 

16951 1799, 1903; 2007' 2110; 

2732 28851 29891 8042 3146^ 

623249 623353'623456' 628559 623668 628766 623869 623978624076 624179 



42S2 43S5! 4488! 4591! 4695 



5812 5415 5518 

63401 6443 6546 

7366! 7468 7571 

8389 1 8491 8598 

94101 9512 9613 



56211 5724 

6648! 6751 

7678; 7775 

8695' 8797 

9715' 9817 



630428 680530 630631 680738 630835 



440 
1 
2 
3 
4 
6 
6 
7 
8 
9 

450 



No. 



1444 1545! 1^4^ 
2457j 2559 2660 

683468 633569 633670 

4477 
5484 
6488 
7490 
8489 
9486 
640481 



1748 1849 

2761! 2862 

633771 688872 

4578' 4679' 4779, 4880 

55841 5685! 5785! 5886 

6588 1 66881 6789 6889 

7590 7690 7790| 7890 

8589! 8689 8789 8888 

9686 9785' 9885 



47 

5827 
6853 

378 



4901 
5929 
6956 

7980 
8900 9002 
9919 630021 
630936 1038 
1951 2052 
2963^ 3064 

633973 634074 
49811 5081 
5986 6087 
6989 7089 
799 a: 8090 
8988 9C88 
9984 640084 



5004; 5107 52101 

6032' 6135 6238 

7058; 7161 7268 

8(82' 8185 8287 

9104i 9206 93(8 
630128 680224 630326 

1139 1241 1342 



2153 
3165 



2255 2356 
3266 8367 



640581 640681)1640779 640879 64(978 10771 1177 



634175 034276 634876 
5182; 5i83 53S3 
61871 6'287 6888 
7189 7290 7390 
8190 8290 8389 
9188 9287 9387' 

640183 640288 64(j882 



1474 1573, 1672 
2465 2563j 2662 

648453 643551643650 



4489 
5422 
6404 
7883 
8360 



4537 1 4636 

5521 ! 5619 

6502' 6600 

7481! 7579 

8458! 8^5'^ 

9432' 9530 



1771; 1871 1970! 2069 2168 
2761 2860 2959, 3058 3156 



643749 643847 

4784 4882 

5717; 5815 

6698, 6796 

7676 7774 

8658' 8750 

9627i 9724 



1276 1875 
2267 2866 
3255 3354 

644242 644340 



643946 644044 644143 

4981: 5029 51271 5226 5324 

5913 6011 1 6110 6208 6806 

6894 6992' 7089 7187 7285 

7872 7969' 8067 8165 8262 

8848 8945 9043' 9140 9287 

9821 9919 650016 650118 65(210 



650308 650405 650502 650599 650696 650798 650890; 0987| 1084 118l| 
1278! ^375! 1472; 1569, 1666! ^762 1859 1956 2053 2150 
2246 23431 2440 2586 2633 2730! 2826' 2923 3019 3116 

653213 653309 653405 653502 653598 653695 653791 658888 653984 0)540801 
4465 4562! 4658 4754' 4850; 4946 6042' 
5427j 5523 6619 5715; 581 o! 5906 6002 
6886 6482 6577| 6673 6769, 6S64 6960: 
7343 7438 7634' 7629 7725 7820 7916 
829S' 8893 8488 8584' 8679 8774 8870 
9250! 9346! 9441 9536: 9681 ! 9726 9821 
9916 660011 660106 660201 660296 660891 660486 660681 660676 661)771 



4177 


4273 


4369 


5138 


5285 


5381 


6u98 


6194 


6290 


7056 


7152 


7247 


8011 


8107 


8202 


8965 


9060 


9155 



660865 
1813 



0960! 1055 
1907 1 2002 

"I 



1150J 1245 
2096, 2191 



1839 1434 1529, 1623 1718; 
2286 2880! 2475 2569 2663 



108 
108 
108 
108 
107 
107 
107 
lOT 
106 
106 

106 
106 
105 
105 
105 
105 
104 
104 
104 
104 

103 
103 
103 
103 
102 
102 
102 
102 
101 
101 

101 
101 
100 
100 
100 
100 
99 
99 
99 
99 

98 



98 
98 
97 
97 
97 
97 
97 

96 
96 
96 
96 
96 
95 
95 
95 
95 
95 



6 I 'O 



8 I 9 I Diff. 





490 








Logarithms 


OF Numbers. 








^0. 

460 


1 1 2 3 


4 


5 1 6 


7 1 8 


9 Diff. 


662753 6G2852 662947'668041 663135 668230 663824 663418 663512' 6686071 94 




1 


3701 


3795 38S9 


1 8988 


4078 


1 4172 4266 4860 4454 


4548 94 




2 


4642 


4786 488( 


1 4924 


5018 


5112 5206 5299 5893 


54871 94 




3 


5581 


5675 5769 


5862 


5956 


6(;50: 6143 62371 6331 


6424! 94 




4 


6518 


6612 6705 


6799 


(892 


6986 7079 7173i 7266 


7860 


94 




6 


7453 


7546 7640 


7788 


7826 


7920 8018 8106; 8199 


8293 


93 




6 


8886 


8479 8572 


8665 


8759 


8852 8945! 9088 9181 


9224 


93 




T 


9317 


9410 9503 


1 9596 


1689 


9782 98761 9967 670060' 670153 


93 




8 


670246 670389 670481' 670524; 67 617'67(m 10 670802 67(895 (988: 1(80 


93 




9 


11781 1265 1358j 1451 


1548 1636 1728 


1621 1918 20(5 


93 




470 


672^98672190 672288 672375 


672467 672560 672652672744 672886'672929' 92 \ 




1 


3'';21 


3113 32(i5: 8297 


8890 8482 8574 


8666 8758 8850! 92 




2 


S942 


4084 4126! 4218 


4810 4402 4494 


4586 46771 47(9! 92 




3 


4861 


4953 5(45! 5187 


5228: 5820 5412 


55(3 6595 5687! 9.2 




4 


5778 


5870 5962 6053, 6145 


6286 6828 


6419 6511 66(21 92 




5 


6C94 


6785 6676; 69(8; 7059 


7151 7242 


7883 7424 7516! 91 




6 


765/7 


76S8 7789 7881 


7972 


8068 8154 


8246 8£86i 8427' 91 




7 


8518 


8609 8700 8791 


S8S2 


8973 9(64 


9155 9246 9^871 91 




8 


9428 


9519' 9610 9700 


9791 


9882 3978 680068 68(154, e8( 245- 91 1 




9 


68u836 680426680517|680607^e80C98 6807S9 680879, (970 1160 1151 91 




480 


681241 '681832681422 681513 '681603 681 (98 681784 681874 681964 6^2155 90 




1 


2145 


2235 2326! 2416' 2506; 2596 2686 


2777 2867 


29571 90 




2 


8047 


3187] 3227 


8317 


8407 


3497 8587 


8677 3767 


S857i 90 




3 


£947 


4037 


i 4127 


4217 


4307 


4?96 4486 


4676 46e6 


4756 90 




4 


4845 


4985 


5025 


5114 


5204 


5294 5883 


5473 5563 


5C52 90 




5 


5742 


5831 


5921 


6010 


6100 


6189 6279 


6868 6458 


6547 89 




6 


6686 


6726 


6815 


6904 


6994 


7088 7172 


7261 7851 


74401 £9 




7 


7529 


7618 


7707 


7796 


7886 


7975 8064 


8153 8242 


8£Slj 89 




8 


8420 


85(9 


8598! 8687 


8776 


8865 8958 


9042 9181 


92201 89 




9 


9309 


9S9S 


9486 9575 


9664 


9753 9841 


9980 690019 


€90107 £9 




490 


G9G196 


690285'690878'690462'e9C55o!(90689C9C728'690S16 6909(5 


€9(C93^ £9 




1 


1C81 


1170i 12581 1347 1485 


1524 1612 


1700 1769 


1877: 88 




2 


1965 


20581 2142! 2280 2818 


2406 2494 


2583 26^1 


27f9 88 




3 


2847 


2985 30231 3111 3199 


8287' 8876 


8463 8551 


8Cr9 88 




4 


3727 


3815 8908 89911 4078 


4166 4254 


4342 448{ 


4517, 88 




5 


4605 


4698 4781 


4868! 4956 


5044 5181 


5219 5807 


5£94 88 




6 


5482 


5569 5657 


5744| 5882 


5919 6(07 


6(94 6182 


('2(9 87 




7 


(856 


6444 


6531 


6618! 6706 


6793; 6880 


€968 7055 


7142 87 




8 


7229 


7317 


7404 


7491 1 7578 


7665: 7752 


78£9 7926 


8014; 87 




9 


8101 


8188 


8275 


8362 8449 


8535 8622 


8709 8796 


8888: 87 




£00 


698970 


699057'699144'69928l'699317'699404'c99491 699578 ef)9e64 


(99751 87 




1 


98881 9924 700011 700098 700184 70(271 700S58 700444 70( f 81 


79« m 87 




2 


700704 7007901 C877i 09631 1050 


1186 1222; 13(9 1£95 


1482 86 




3 


1568 


1654 


1741 18271 1918 


1999 2086! 2172 2258 


2844 86 




4 


2481 


2517 


2603 2689 2775 


2861! 2947i 8(83 3119 


821 5 86 




6 


8291 


3377 


3468 8549 86£5 8721! £807! 3893 £979 


4(65 86 




6 


4151 


4286 


4322 4408 4494 


4579 4665 4751 4887 


4922 86 




7 


50(3 


5094 


5179; 5265 


5350 


6486' 55221 5607! 5698 


5778 86 




8 


5864 


5949 


6085 6120 


6206 


6291 6376' 6462' 6547 


6682 85 




9 


6718 


6808 


6888 6974 


7059 


7144, 7229 1 7315; 7400 


7485j 85 




510 


707570 707655707740 707826'70791l'707996 7(8(81 7C8166'7(8251 


7(8886' 85 




1 


84211 


8506: 85911 8676i 87611 8846 89811 9015 9100 


9185! 85 


! 


2 


9270l 


9855 94401 9524 96091 9694 9779 9863 9948 


710(83; 85 




3 


710117 710202 710287 '^10871 710456 710640 710625 710710 710794 


(879 85 




4 


C963 


1048 1132! 12171 13011 


1885! 1470 


K54 16£9 


1723; 84 




5 


1807 


1892 


1976; 2060 2144 


2229, 2313 


2897 2481 


2566 84 




6 


2650 


2734 


2818' 29G2; 2986i 


3070 


3154 


3288 8323 


3407 84 




7 


3491 


3575 


3659! 8742 8826 


8910 


3994 


4078 4162 


4246 84 


j 


8 


4330 


4414 


4497! 45811 4665 


4749 


4883 4916 5(00 1 


6(84! 84 




9 


5167 


5251 


5335| 5418! 55C2 


5586 


5669 5753 5886 


5920' 84 







1 


2 1 


3 1 4 1 


5 


6 1 7 1 8 


y \Dim 



Logarithms of Lumbers. 



491 



No. 





1 


2 


3 


4 


5 


6 


1 


8 


9 


Diff. 


520 


716')03'71608T 716170 716254 


716387 716421 716504 716588; 716671 716754; 83 i 


1 


68BS 


6921 7004 7088 


71711 7254! 7338 7421 


7504! 7587! 83 


2 


7671 


7754 7837( 7920 


8003 8086 


8169, 8-263 


8336i 8419: 83 


3 


8502 


8585 8668 8751 


8834' 8917 


9000 9ii83 


9165' 9248i 83 


4 


9331 


9414! 9497. 9580 


9663' 9745! 9823' 9911 


9994 7200771 83 


5 


720159 


720242 720325 720407 


720490 720573 720655 720738 


720821! 0003; 83 


6 


0936 


1068 1151! 1233 


1316 1398; 1481 1563 


1646; 1728' 82 


7 


1811 


1893 1975 2058 


2140, 22221 2305 2387 


2469! 2552 82 


8 


2634 


2716 2798 2831 


2983 3045 3127 3209 


3291 


8374 82 


9 


3456 


3538 3820 37o2 


3784 3866 8948 403 J 4112 


4194 82 


630 


724276 


724358 724440 724522 


724604724685 724767 724849'72493l'725013l 82 I 


t 


5095 


51761 5258: 5340 


5422 


5503 5585, 5667| 5748 


68301 82 


2 


5912 


5993; 6075' 6156 


6238! 


6320 6401 6483! 6564 


6646! 82 


3 


672- 


6809; 6890 6972 


7053! 


7134! 72161 7297 


7379 


74601 81 


4 


7541 


7623 77;)4' 7785 


7866 


7948! 8029 8110 


8191 


8278 81 


5 


8354 


8435; 8516 8597 


8678 


8759; 8841 8922 


90O3 


9084' 81 


6 


9165 


92461 9327i 9408 


9489 


9570! 9651 9732 


9813 


9393 81 


7 


9974 


730055 73J136 730217 


730298 


730378 730459 730540 730621 


T3i*7i2 81 


8 


7307S2 


0863 


0944, 1024 


1106, 


1186 1266! 1347 1423 


1508' 81 


9 


1589 


1669 


1750 1880 


1911 


1991 2072| 2152 2238 


2313 81 


540 


732B94 


732474 


732555 782635 


732715 


r32796 732876 732956 738037 


783117 80 


1 


3197 


3278 


8358; 8438 


3518 


3598' 8679 3759; 3839 


3919! 80 


2 


3999 


4079 


4160 4240 


43201 


4400 


4480 4560 4640 


4720 80 


3 


4300 


4880 


4960 5040 


51201 5200 


5279! 6359 6489 


6519 80 


4 


5599 


5679 


5759 5333 


5918 5998 


6078! 6157 6237 


63171 80 


5 


6397 


6476 6556; 6635 


6715| 6795 6874! 6954J 7084 


7118; 80 


6 


7193 


7272 7352 7431 


7511 7590I 7670' 7749 7829 


7908 79 


7 


7987 


8067: 8146 8225 


8305 83841 8463 8543 8622 


87011 79 


8 


8781 


8880 8939 


9a8 9097 9177! 9256 9335 94141 9493, 79 


9 


9572 


9651; 9731 


9810, 9339 996S740047|740126,740205;740284! 79 


550 


740363 


740442 740521 740600 740678' 740757 740836 7409151 740994'741 073 


79 


1 


1152 


1230 1309; 1388 14671 15461 1624 


1708 


1782 


1860 


79 


2 


1939 


2018 2096 2175 2254| 2332i 2411 


2489 


2568 


2647 


79 


3 


2725 


2804 2382 


2961 8039 3118; 3196 


8275 


3858 


3481 


78 


4 


8510 


3533 3667 


8745 8823 


8902; 8930 


4058 


4136 


4215 


78 


5 


4298 


4371 4449 


4523 4606 


4684 4762 


4840 


4919 


4997 


78 


6 


5075 


5153 5231 


5309 5387 


5465 5543 


5621 


6699 


5777 


78 


7 


5855 


5933 6011 


6089 6167 


6245 6323 


6401 


6479 


6556 


78 


8 


6634 


6712 6790 


6883 6945 7028; 7101 


7179 


7256 


7834 78 ! 


9 


7412 


7489_ 7567 


7645 7722 7800 7878 


7955 


8033 


81101 78 1 


560 


748188 


748266 748343 74842l'748498'748576'74S653'748731 


748808 748835! 77 1 


1 


8963 


9)40 9118 9195 9272 935o! 94271 9504 


9582! 9659 77 


2 


9736 


9814 9S91i 9988 750045 750123 750200 750277,750354 7504yii 77 1 


3 


7505 J8 


750586 750663 750740 0817 0894; 0971 


1048 


1125 


1202, 77 


4 


1279 


1356 1433, 1510 1587 


1664! 1741 


1818 


1895 


1972 77 


5 


2048 


2125 2202 2279 2356 


2433! 2509 


2586 


2663 


2740; 77 


6 


2816 


2S93 2970 3047 8123 3200| 8277 


3353 


8480 


3506, 77 


7 


3583 


3660 3786 8813 8889, 8966' 4042 


4119 4195 


4272 77 


8 


4348 


4425 4501 4578 4654 4730 ; 4807 


4883 4960 


5086; 76 


9 


5112 


5189^ 5265 5341^ 5417 5494 5570 


. 6646 6722 


5799, 76 


570 


755875 


755951 756027 756103 756180 756256 756332'756408'756484'756560| 76 1 


1 


6036 


6712 6788 6864 6940 70161 7092! 71681 7244 


78201 76 


2 


7396 


7472 7548 7624 7700 7775| 7851 7927 8003 


8079, 76 


8 


8155 


8230 8306 8382 8453 8583' 8609 8685 8761 


8886 76 


4 


8912 


8988 9063 9139 9214' 9290| 9366! 944ll 95171 9692 76 


5 


9668 


9743 9319| 9S94 9970 760045 760121760196 760272 760847, 75 


6 


760422 


760498 700573 760649 760724 0799 0875 


C950 


10251 1101 75 


7 


1176 


1251, 132S 


1402 1477 


1552 1627 


1702 


1778! 1858: 75 


8 


1928 


2003, 2078 


2153 2228 


2303 2878 


2453 


2529; 2604; 75 


9 

No. 


2679 


2754 2829 


2904 2973 


3053 8128 


8203 


8278 8858 


75 


1 


2 


~3~r'"4'"' 


yr 6"^ 


7 


"8 9~ 


Diff. 



492 








LoGARITnMS 


OF iS 


^UMBERS. 








No. 


! 1 


2 


3-456 


7 8 9 |DifF. 


580 


763428 763503 763578763658 763727 763802 763877 763952 764027 764101 75 


1 


4176 4251 


4326 


4400. 4475! 4550 4624; 4699 4774= 4848 75 


2 


4923 4998 


5072 


5147 5221 


5296: 53701 5445 5520' 5594 75 


3 


5669 5748 


5818 


5892 5966 


6041j 6115i 6190: 6264' 6338 74 


4 


6413 6487 


6562 


6636 6710 


6785! 6859 6933| 7007! 7082 74 
7527| 7601 7675: 7749, 7823 74 


5 


7156 7230 


7304 


7379, 7453 


6 


7898 7972 


8046 


8120 8194 


8268 8342 8416 84901 8564' 74 


7 


8638 8712 


8786 


8860: 8934 


9008 9082 9156' 92301 9303; 74 


8 


9377 9451 


9525 


9599 9673 


9746' 9820 9894' 9968 770042: 74 


9 


770115 770189 770263 


770336 770410 


770484 770557 770631770705 0778 74 


590 


770852 770926 


770999 


771073 771146 


771220 771293'771367 771440 7715141 74 


1 


1587 1661 


1734 


1808 1381 


1955 2.(28 2102 2175; 2248' 73 


! 2 


2322 2395 


2468 


2542 2615 


2688' 2762 


2835' 29081 2981 


73 


3 


3055 3128 


3201 


3274 3348 


3421 1 3494 


3567i 3640| 3713 


73 


4 


3786 3860 


3933 


4006 4079 


4152 4225 


4298 4371 4444 


73 


5 


4517 4590 


4663 


4736 4809 


4882 4955 


5028' 5100 5173 


73 


6 


5246 5319 


5892 


5465 5538 


5610l 5683 


5756: 5829 5902 


73 


T 


5974 6047 


6120 


6193 6265 


6388 6411 


6483 6556 6629 


73 


8 


6701 6774 


6846 


6919 6992 


7064' 7137 


7209; 7282 7354 


73 


9 


7427 7499 


7572 


7644 7717 


7789| 7862 


7934 8006 8079 


72 


600 778151 TT8224 


778296 


778368 778441 


778513 778585 778658' 778730 778802 


72 


1 8874 8947 


9019 


9t'91 916r 


9236 9308 9380: 9452 9524 


72 


2 9596 9869 


9741 


9813 938t. 


9957 780029 780101 780173 780245 


72 


3 780317 780389 


780461 


780533 7S06U5 78;J677i 0749, 0821i 0893, 0965 


72 


4 


1037 1109 


1181 


1253 1824 1396 1468 


1540 I6I2: 1684 


72 


6 


1755 1827 


1899 


197l! 20421 2114' 2186 


2258 2329, 2401 


72 


6 


2473, 2544 


2616 


2688, 2759: 2831 2902 


2974 


3046 3117 


72 


7 


3189 3260 


3332 


34! ^3 3475j 3546 3618 


3689 


3761 


3832 


71 


8 


3904 


3975 


4046 


4118 4189 4261 4832 


4403 


4475 


4546 


71 


9 


4617 


4689 


4760 


4831 49U2, 4974 5045 


5116 


5187 


5259 


71 


610 


785330 


785401 


785472 


785543 785615 785686 785757 785828785899785970 


71 


1 


6041 


6112 


6183 


6254 


6325: 6396: ^467 


6538 


66091 6680 


71 


2 


6751 


6822 


6893 


6964 


7035 7106 7177 


7248 


7319: 7390 


71 


3 


7460 


7531 


7602 


7673 


7744 7815 7885 


7956 


8027i 8i;98 


71 


4 


8168 


8239 


8810 


8381 


8451 8522 8593 


8663 


8734i 8804 


71 


5 


8875 


8946 


9016 


9087 


9157 9228' 9299' 9369' 9440' 9510 


71 


6 


9581 9651 


9722 


9792 9363' 9933 790004 790074 790144 790215 


70 


7 790285:790356 


790426 


790496 790567 790637 


0707; 0778 


0848 0918 


70 


8 


0988 


1059 


1129 


1199, 1269 1340 


1410: 1480 


1550! 1620 


70 


9 


1691 


1761 


1831 


1901 1971! 2041 


2111 2181 


2252 2322 


70 


620 792892', 792462 


T92532 792602'792672'792742'792812 792882792952 793022 


70 


1 


3092 


3162 


3231 


33011 3371i 3441 


3511. 85811 36511 3721 


70 


2 


3790 


3860 


3930 


40001 4070! 4139 


4209; 4279 4349, 4418 


70 


3 


4488 


4558 


4627 


4697! 4787i 4836 


4906; 4976, 5045 5115 


70 


4 


5185 


5254 


5324 


53931 5463 5532 


5602; 5672: 5741 5811 


70 


5 


5S80 


5949 


6019 


6088 6158 6227 


62971 6366! 6436 6505 


69 


6 


6574 


6644 


6713 


6782 6852 6921 


69901 7060 


7129 7198 


69 


7 


7268 


7337 


7406 


7475 7545 7614 


7683J 7752 


7821 7890 


69 


8 


7960 


8029 


8098 


81671 8236 8305 
8858 8927j 8996 


8374 8443 


8513 8582 


69 


9 


8651 


8720 


8789 


9065 9134| 92031 9272 


69 


j 630 799341 


799409 


799478 


799547 799616799685 799754'799823'799892'799961 


69 


1 800029:800098 


800167 


800236 800305 300373 800442 800511 800580 800648 


69 


2 


0717 


0786 


0854 


0923, C992 IO6I1 1129, 1198! 1266 1335 


69 


3 


1404 


1472 


1541 


1609 


1678 1747 1815; 1884 


1952 2021 


69 


4 


2089 


2158 


2226 


2295 


2363' 2432 2500 2568 


2637 


2705 


68 


5 


2774 


2842 


2910 


2979 


3047i 3116' 3184' 3252 


3321 


3389 


68 


6 


3457 


3525 


3594 


3662 


3730; 3798' 3867! 8935 


4003 


4071 


68 


7 


4139 


4208 


4276 


4344 


4412 4480 


4548 4616 


4685 


4753 


68 


8 


4821 


4889 


4957 


5025 


6093 5161 


5229: 5297 


5365 


6433 


68 


9 


5501 


5569 


5637 


5705 


5773, 5841 


5908 5976 


6044 


6112 


68 


No. 





1 


2 


3 


4 1 5 


6 1 7 


8 


9 


Diff. 



Logarithms of Kumbers. 



493 



No. 





1 2 3 


4 5 6 


1 1 8 


9 


Diff. 


640 


806180'806248 806816'806384 


806451 8C6519:80e587:806655,8S6723 806790 


68 


1 


6858 


6926 6994 


7J61 


7129 7197 


7264 


7332 


7400 


7467 


68 


2 


7535 


7603 7670 


77 8 


78061 7873 


7941 


8008 


8076 


8143 


68 


3 


8211 


8279' 8346 


84H 


8481 : 8549 


8616 


8684 


8751 


8818 


67 


4 


8886 


8953i 9021 


908^- 


9156; 9223 


9290 


9358 


9425 


9492 


67 


5 


9560 


96271 9694 


9762 


9829: 9896 


9964 


810031 810098 810165 


67 


6 


810233:810300:81(337 810434:810501 810569:810636 


07031 0770 


C887 


67 


7 


0904 


0971 1039 


1106! 1173 


1240 


1807 


1374| 1441 


1508 


67 


8 


1575 


1642 1709 


1776 


1843 


1910 


1977 


2044 2111 


2178 


67 


9 


2245 


2312 2379 


2445 


2512 2579 


2646 


2713 2780 


2847 


67 


650 


812913'812980 


813047 


813114 813181 Sl3247|813314!81338l'813448'813514 


67 


1 


3581 


3648 


3714 


3781 


3848 8914 


8981 


40481 4114! 4181 


67 


2 


4248 


4314 


4381 


4447 


4514 4581 


4647 


4714 


4780 


4847 


67 


3 


4913 


4980 


5046 


5113 


5179 5246 


5312 


5878 


5445 


5511 


66 


4 


5578 


5644 


5711 


5777 


5843: 5910 


5976 


6042 


6109 


6175 


66 


5 


6241 


6308 


6374 


6440 


6506 i 6573 


6689 


6705 


6771 


6838 


66 


6 


6904 


6970 


7036 


7102 


7169: 7285 


7301 


7867 


7438 


7499 


66 1 


7 


7565 


7631 


7698 


7764 


78301 7896 


7962 


8028 


8C94 


8160 


66 


8 


8226 


8292 


8358 


8424 


8490' 8556 


8622 


8688 


8754 


8820 


66 


9 


8885 


8951 


9017 


9083 


9149, 9215 


9281 


_9346 


9412 


9478 


66 


660 


819544 819610^819676 


819741 


819807 819878 81 9989'82CCr4'&20G70 820186 


66 


1 820201'820267;820333 820S99'820464 82C530'&2:.595 


C661 0727 


0792 


66 


2 


0858 


0924 


0989 


1055 


1120 


1186: 1251 


1817 1882- 


1448 


66 


3 


1514 


1579 


1645 


1710 


1775 


1841 ! 1906 


1972 2037 


2103 


65 1 


4 


2168 


2233 


2299 


2364 


2480 


2495 2560 


2626t 2691 


2756 


65 ! 


5 


2822 


2887 


2952 


3C18 


3(83 


3148 8213 


8279, 8344 


84(9 


66 1 


6 


3474 


3539 


3605 


8670 


3735 


8800 


8865 


898o! 8996 


4061 


65 ! 


7 


4126 


4191 


4256 


4321 


4386 


4451 


4516 


4581 


4646 


4711 


65 1 


8 


4776 


4841 


4906 


4971 1 5036 


5101 


5166 


5231 


5296! 5861 


65 1 


9 


5426 


5491 


5556 


5621 5686 


5761 


5615 


5880 


£945 6010 


65 1 


670 


826075 826140 


826204 


826269 826834 826g99's26464 


826528 82G593'826658 


65 1 


1 


6723 


6787 


6852 


6917: 6981 


7046 


7111 


7175 


7240 


7805 


65 


2 


7369 


7434 


7499 


7563, 7628 


7692 


7757 


7821 


7686 


7951 


65 ! 


3 


8015 


8080 


8144 


8209; 8278 


8338 


84C2 8467 


8581 


8595 


64 ! 


4 


8660 


8724 


8789 


8853' 8918 


8982 


9046: 9111 


9175 


9289 


64 i 


5 


9304 


9368 9432 


9497; 9561 


9625 


9G90: 9754 


9818 9882 


64 


6 


9947 830011830075 


830189 830204 


S30268 83(882 830896 83C460:880525 


64 i 


7 


830589 


0653 


0717 


0781 i €845 


C9G9 


0973 1087 


1102 1166 


64 1 


8 


1230 


1294 


1358 


1422! I486 


1550 


1614! 1678 


1742 1806 


64 i 


9 


1870 


1934 


1998 


2062 2126 


2189 


2258, 2317 


2881 2445 


64 ! 


680 


832509 832573832637 


83270o'882764 


832828832892 832956'833C20!883C83 


64 


1 


3147 3211 


3275 


3338; 3402 


3466 


35801 8593 


8657 


3721 


64 


2 


3784 3848 


3912 


8975! 4039 


4103 


4166 4230 


4294 


4857 


64 


8 


4421 4484 


4548 


4611! 4675 


4789 


48021 4866 


4929 


4998 


64 i 


4 


5056, 5120 


5183 


5247 


5810 


5373 


5437 5500 


5564 


5627 


G3 i 


6 


5691 5754 


5817 


5881 


5944 


6007 


6071 6134 


6197 


6261 


63 


6 


6324' 6387 


6451 


6514 


6577 


6641 


6704 6767 


6880 


6894 


68 


7 


69571 7020 


7083 


7146 7210 


7273 


7336: 7899 


7462 


7525 


68 


8 


7588 7652 


7715 


7778^ 7841 


7904 


7967| 8080 


8C93 


8156 


68 


9 


8219| 8282 


8345 


84C8 8471 


8534 


8597| 8660 


8723 


8786 


68 


690 


838849 888912838975 


839088 839101 


889164 


889227 8892S9'889852'8S9415 


68 


1 


9478 9541 1 9604 


9667 9729 


9792 


9855 9918! 99811840043 


68 


2 


840106 840169 840232 


840294 840857 


840420 


840482 840545 840608 


0671 


63 


8 


0733, 0796 


0859 


0921 (984 


1046 


1109 11721 1234 


1297 


63 


4 


1359, 1422 


1485 


1547 1010 


1672 


17351 1797 


1860 


1922 


63 


5 


1985' 2047 


2110 


2172 2235 


2297 


2360 2422 


2484 


2547 


62 


6 


2609, 2672 


2734 


27961 2859 


2921 


2983' 8046 


8108 


8170 


, 62 


7 


8233, 8295 


8357 


8420 


8482 


3544 


3606, 8669 


8731 


8793 


62 


8 


8855 8918 


8980 


4042 


4104 


4166 


4229' 4291 


4858 


4415 


62 


9 


4477 4689 


4601 


4664 


4726 


4788 
6 


4850 
6 


4912 


4974 


5086 


i 62 
1 Diff. 


No. 1 1 1 


2 


3 


4 


'> 


8 



42 



494 



Logarithms of Numbers. 



JSTo. i ; 1 2 1 3 1 4 5 1 6 1 7 1 8 1 9 


Diff. 


700 


84509S 815160 845222 S4528 4 845340 845408 845470 845532 S45594 845S56' 


62 


1 


5713 578) 58t-2 


59J4 5936 6028 6u99 6151 6218 6275 


62 


2 


6887 6399 6461 


6528 6585 6646 67u8 6770 6882 6894: 


62 


3 


6955 7017, 7079 


7141 7202 7264 7826 7883 7449 751 1! 


62 


4 


7578 7634 7693 


7753 7819 7881 7948 8904 8j63 8128 


62 


5 


8189 8251 8312 


8874 8435 8497 8559 8629 8682 8743 


62 


6 


1 83 j5 831^3 8928 


. 8939 9951 9112 9174 9285 9297 9358' 


61 


7 


' 9419 9431 9542 


■ 96)4 9365 9726 9783 9849 9911 9972 


61 


8 


850083 35J095 850156 850217 850279 850340 85u4j1 850462 850524 S50585 


61 


9 


i 0640 0707| 0769; 0880^ 0891 0952 1014 1075 1136 1197 


61 


710 


;8512-)8S51320 851881 851442 851503 S515G4 851625 851686 851747 S51309 


61 


1 


! ISro 1931 


1992, 2058 2114 2175 2286 2297 2858 2419 


61 


2 


2183! 2541 


2602 2633 2724 2735 2846 2907 2933 3j29; 


61 ■ 


3 


3.)0J^ 3153 


3211 3272 3338 3394 3455 3516 3577 3637! 


61 


4 


3o93 3759 


3320 3331 3941 4002 4068 4124 4185 4245 


61 


5 


43J0: 4837 


4423! 443S 4549 4610 4670 4731 4792 4352 


61 


6 


4918' 4974 


5084! 5095 5153 5216 5277 5337 5393 5459' 


61 


7 


5519 5530 


58401 5701 5761 5822 5332 5943 6003 6004 


61 


8 


6124 6185 


6245' 6396 6866 6427; 6437; 6548 6603 C638; 


60 


9 


6729 6789 


6850' 6919; 6970 7031! 7091^ 7152 7212 7272! 


60 


723 


357832 857898 857458 S57513 857574' 857684 85 7694857755 857815 857875; 


60 


1 


7985 7995 


8056 8116! 8176 8286 8297 8857 8417 8477' 


60 


2 


8587 8597 


8057 


8718; 8773 8338 8898 8953 9018 9o78; 


60 


3 


9183 9193 9253 


9318 9879 9439 9499 9559 9319 9379' 


60 


4 


9789 9799 9359 9918 9973 36J033 360093 36 .'153 S6u21S 86j278' 


60 


5 


33)883 33J393 36)453 86 J51SS30573 0687 0697 0757 0817 0877; 


60 


6 


0037 0993 1053 


1116 1176 1286 1295 1355 1415 1475 


60 


7 


1534 1594 1654 


1714 1778 1833 1893 1952 2012 21)72^ 


60 


8 


2181^ 2191 2251 


2810: 2870 2430 2489 2549 2608 2668 


60 


9 


2723! 273 7| 2847 


2996, 2933 3025 3985 3144 3204 3263 


60 


730 


S38328-833332'533442 8635D1 863531 833820 868680'863739 863799 863358^ 


59 


1 


3917 3977^ 4)83i 4093 4155 4214 4274 4^88 4892 4452 


59 


2 


4511 4570 46301 4689 4743 4308 4367 4923 4985 5045 


59 


3 


5104 5168 5222' 5232' 5341 549 J 5459 5519 5573 5637 


59 


4 


53931 5755 5314^ 5374* 5938 5992 6051 6110 6169 6228 


59 


5 


6237, 6843 6405 6435. 6524 6538 6642 6701 6760 6819 


59 


6 


6378 6037 6993; 7055' 7114 7178 7232 7291 7350 7439 


59 


7 


74371 7523 7535' 7644 -7708 7762 7821 7330 7939 7993 


59 


8 


8958, 8115 8174 8288, 8292 8350 84 j9 8468 8527 8536 


59 


9 


8644' 8703 8732^ 8321; 8379 8938 8997 9056 9114 9173 


59 


740 


33a232's69299S69349's39408's39436 839525869584 S69642 839701 8G9760' 


59 


1 


9313 9377 9935 9994 370058 370111 870170 870228 870237 8705345 


59 


2 


870404 870432 370521870579, 0683 0693 0755 0818 0872 0980 


58 


3 


0939 1047 1106: 1164 1228 1231 1839 1393 1456 1515 


58 


4 


1578 1631 1699' 1748 1896 1865 1928 1931 2040 2098 


58 


6 


2153 2215 2273 2831 i 2389 2443 2506 2564 2622 2681 


58 


6 


2T39 2797 2355 2913 2972 3030 3088 3146 3204' 3262 


58 


7 


3321 3879 3487, 3495; 3553 3611 3669 3727 3785; 3844 


58 


8 


3992, 3939, 4013 4976 4134 4192 4250! 4308 4366' 4424 


58 


9 


4432^ 4549^ 4593^ 4656! 4714 4772, 4830, 4838' 4945 5003 


58 


750 


875D61 S75119 375177 S75285 375293375351 875409 S75466's75524's755S2 


58 


1 


5649' 5393 5756 


5313, 5371 5929 5937 6045 6102 616;; 


58 


2 


6218 6276 6833 


6391 6449 6597 6564 6622 6689 6737 


58 


3 


6795 6853 6910 


6908, 7026 7083 7141 7199, 7256 7314 


58 


4 


7371 7439 7437 


7544! 7602 7659 7717 7774 7882 7859 


58 


5 


7947 8004 8962' 


8119; 8177 8234 8292 8349 8407 8464 


57 


6 


8522 8579 8637 


8694; 8752 8839 8366 8924 8981 9039 


57 


7 


9093 9153 9211J 


9263' 9325 9383 9440 9497 9555 9612 


57 


8 


9869 9723 97S4 


9841 9893 9956 380013 S80070 889127 S80185 


57 


9 

No. 


889242 889299 880353 830413 380471 880523 0585 0642 0699 0756. 


57 


0;li2i3i4i5i67i8|9 i Diff. { 



Logarithms of Numbers. 



495 



No. 



760 
1 
2 
3 
4 
5 
6 
7 
8 
9 

770 
1 
2 
8 
4 
5 
6 
7 
8 
9 

780 
1 
2 
3 
4 
5 
6 
7 



790 
1 
2 
3 
4 
5 



800 
1 

2 
8 
4 
5 
C 
7 
8 
9 

810 
1 
2 
8 
4 
5 
6 
7 
8 
9^ 

No. 







1 



9 iDiff. 



880814 88C8T1 

J 385: 1442, 

1955 2012 

2525' 2581 

81 93 8150, 

3661 3718 

4229: 4285 

4795 4852! 

5361 5418 

5926' 59S3| 

886491880547 

7054: 71111 

7617i 7674' 

8179: 8286' 

8741 8797 

9302 9858 

9862 9918' 

890421 890477 

09301 1035; 

1537| 1593 



880928 88G9S5 881042 881G99 
1499; 1556 



2069 
2688 
82i)7 
8775 



2126 
2695 
8264 

., 8832 
4842 4899 
49i9 4965 
5474; 5581! 
6089 6096 

880604 886660 
71671 7223 
7780 7786 
8292, 8848 
8853: 89.9 
94I4I 9470 
9974 89o030 

8905881 0589 
1(911 1147 
1649! 1705 



1613 


1670 


2183 


2240 


2752 


2809 


8821 


8877 


8888 


8945 


4455 


4512 


5022 


5078 


5587 


5644 


6152 


6209 



881156 831 21 3 

1727, 1784| 

2297| 2854 

2866 2923 

8434' 8491 : 

4002 4059: 

4569, 4625' 

5185J 5192, 

5700 5757 1 

6265J 6321 



886716 88677; 

7280J 7836' 

7S42| 7898, 

8404: 8460 

8965 9021 

9526! 9582: 
890086 89.141 890197 89^253 



5 886829 
7892 
7955 
8516 
9077 
9638 



886885 
7449 
8011! 
85781 
9134 
9694 



0645 
12!8 
1760 



07001 
1259! 
18161 



0756 
1314 

1872 



0812 

1370 

r 1928, 



831271 8B132S 57 

18411 1898' 57 

241l! 2468! 57 

298}| 3037! 57 

3548 3605 57 

4115' 4172; 57 

4682' 4739! ^'^ 

5248! 5305 57 

5813 5870 57 

6378 6484 58 

886942 886993: 56 

75f:5i 7501 56 

8067] 8128, 56 

8629 8685 56 

9190| 9246: 56 

97501 98061 56 

89081 9 890865' 56 

0868' 0924: 56 

1426 1482! 56 

1983 2C89 56 



892G95 892150,892206 892262 892317 892878 892429,892484 892540 892595 



2651 

8207 
3762 
4316 

4870 
5428 
5975 
6526 

7077 1 



2707; 
3262- 
8817 
4871 1 
4925' 
5478' 
6G30| 
65811 
7182J 



2762 
8318 

8873 
4427 
4980 
5533; 
6085! 
66861 
71871 



2818 
8878 

8928 
4482 
5(!86 
5588 
61401 
6692 
7242' 



3429 
3934 
4588 
5091 
5644 
6195 
6747 
7297 



2929! 

84S4| 
4089; 
4598 
5146 
5699 
6251; 
6802 
7852 



2985 

8540| 
4(94' 

4648! 
5201] 

5754! 
63(6! 

6S57i 
74{;7; 



3040! 
8595 
4150' 
4704' 
5257i 
58C9: 
68611 
6912! 
7462 



3096 8151 

8651 3706 

4205 4261 

4759 4814 

5312 53671 



5864 5920 

6416 6471 1 

6967 7022; 

7517 7572 



897627 897682 897737 897792 897847 8979(2 897957 898ol2 898067 898122 
81761 8281' 8286, 8841, 8:]96| 8451 8506, 8561! 8615 8670 
8725 8780: 8885 8390" &944| 8999 9054 91(9 9164 9218 
9273! 9328' 9388! 9487 9492i 9547 962 9656 9711 9766! 
9821' 9875' 99301 9985 900089 9a0( 94 9G0149 9002(8 900258 900312 



900367 900422 900476 900531 ; 0586 

0913| 0968, 1022 l077i 1131 

1458' 1513 1567! 1622 1676 

2003| 2057 2112 2166 2221 

2547 2601; 2655! 271o! 2764 



0640 
1186 
1781 

2275 

2818 



0095 
1240! 
1785' 
2829 
2873 



07-^9 C804 C859: 

1295 1849 1404! 

1840; 1^94 1948; 

2884' 2488 2492, 

2927j 2081 8086; 



903090 933144 903199 908258 903307 903801 9C3416 903470 903524 903578 



3633; 
41741 
4716 
5256i 
5796' 
6385! 
6874 
74111 
7949 



3687: 


8741 


4229 


4283 


4770 


4824 


5810| 


5864 


5850; 


5904 


68S9 


6448 


6927! 


6931 ! 


7465 


7519! 


8002 


8056! 



8795 
4887! 
4878 
5418 
5958' 
6497: 
7(t85' 
7578, 
8110| 



S849| 

4891 
4982 
5472 
6012 
(■551 
7(i8D 
7626 
8163 



8904 
4445 
498G 
5526 
6066 
6604' 
7143 
7680 
8217 



8958 
4499 

5040! 
5580l 
6119 
6658 
7196 
7784 
82T0 



4012' 
4558; 
6t94l 
5634 
6178; 
6712! 
7250 
7787 
8324 



4066 
4607 
5148 
5688 
6227 
6766 
7304 
7841 
8378 



4120 
4661 
5202, 
5742! 
6281 1 
6820; 
7858 
7895; 
8481 



908485 9;i8589 908592 908646 908699 908758 908807 9:8860 908014 908967 

901211 9074 9123, 9181, 9235| 9289 9842 98J>6 9449 9503! 

I 9556' 9610 9668 9716 9770| 9823 9877 9980 9934 910087 

910091 910144 910197 910251 910304 910858 910411 910464 910518 0571 



0024| 0678 0731 

1158 1211 1264 

1691) 1743 1797 

2222 2275 2828 

2758 2806 2859 

8337| 8390 







1 



0784 
1817; 
1850 
2831 1 
2913 
8443 

"' 8 I " 



0888 
1371 
1908 
2485 
2966 
8496 



0891 1 
1424 
1956 
2488 
8019 
3549 



C944 


0998, 


1477 


158,0 


2009 


2063 


2541 


2594 


3072 


8125 


8602 


8656 


() 


7 



1051 
1584 
2116 
2647 
8178 
8708 



1104 
1687 
2169 
2700 
8231 
8761 



56 
56 
56 
55 
55 
55 
55 
55 
55 
55 

55 

55 
55 
55 
55 
55 
55 
54 
54 
54 

54 
54 
54 
54 
54 
54 
54 
54 
54 
54 

54 
54 
53 
58 
58 
53 
58 
58 
58 
58 

Diff, 



496 



Logarithms of Numbers. 



No. 





1 ! 2 


3 


4 


5 6 1 7 1 8 


9 1 Diff. 


820 ;913814 913867 913920 91S973 914026 914079 9141G2 91 41 84 914237 914290 53 


1 


4343 


4396 4449, 4502 4555 


4608 4660; 4713 4766 


, 4819 58 


2 


4872 


4925 4977; 5030! 5083 


5186 5189! 5241 5294 


5347 53 


3 


54C0 


5458 5505; 5558 5611 


5664 5716! 57C9 5822 


; 5875! 53 


4 


5927 


5980 60331 6G85 6138 


6191 6243] 6296 6349 


i 6401 53 


5 


6454 


6507, 6559, 6612 6664 


6717 6770 6822 6875 


1 6927 53 


6 


6980 


7033. 7085 7188 7190 


7243 7295 7848 74(( 


1 7453 53 


7 


7506 


7558 7611 7663! 7716 


7768 7820; 7873 7925 


; 7978 52 


8 


8030 


8083: 8135 8188 8240 


8298 8845' 8397 845v 


8502 52 


9 


8555 


8607| 8(559 8712; 8764 


h816 8&('9 8921 8978 


9(26 32 


830 '919078 


919130 919183 919235 91 9287* 919r40 919892 919444 91949c'91£549 52 1 


1,1 960li 9653 9706 9758 9810! 9 62 9914 9967 921(19 920(71 52 1 


2 920123 920176 920228 9202SO 92( 3£2 920 84 920486 920489 C541 


; 0598 52 


3 


0645, 0697 0749, 0801; C853 


0SC6 (958 1010 1062 


1114 52 


4 


1166 


1218 1270 


1822 1374 


1426 1478 1580 15S2 


: 1684 52 


5 


1686 


1788 1790 


1S42! 1894 


1946 1998 2(50 2102 


2154 52 


6 


2206' 2258 2810 


2362^ 2414 


2466 2518 2570 2622 


2674 52 


7 


2725 27771 2829: 288i; 2983 


2985 8(87 3(89 8140 


3192 52 


8 


3244 3296' 8848 i £899. 8451 


£5(3 8555 86(^7 8658 


3710 52 


9 1 3762 3814, 8865j £917 B9C9'^ 4C21 4072 4124 4176 


4228 52 


840 '924279 924381924883 924434 924486 924588 924589 924641 924C98 


924744' 52 


1 


4796 


4848 4899; 4951, 50(3 


1 5054 5106 5157 52(9 


: 5261 52 


2 


5312 


5364' 5415 


5467 


i 5518 


5570 5621! 5673 5725 


5776 52 


3 


5828 


6879 5931 


5982 


: 6034 


6085 6187 6188' 6240 


6291 51 


4 


6342 


6894 6445 


6497 


j 6548 


6600 6051 6702 6754 


6805 51 


5 


6857 


C9C8 69591 7011 


7062 


7114 7165 7216 7268 


7819 51 


6 


7370 


7422 7473 


7524 


! 7576 


7627 7678 7780 7781 


7882 51 


7 


7883 


79851 7986 


8(37 


i 8C88 


8140 8191 8242 8298 


8345 51 


8 


8396 


8447i 8498 


8549 


8601 


8652 8708 8754 8805 


8857 61 


9 


8908 


8959, 9010 


9C61 


9112 


9163 9215 9266 9317 


9868 51 


i 850 


929419 929470929521 '929572929628929674 929725 929776 929827 


9S9879; 51 


1 


9980! 9981 930032 930088 930184 980185 98( 286 980287 930888 


930889 51 


2 930440 9804911 C542; (592 


0643, CC94 0745 0796 (847 


0898 51 


3 


0949 


lOOOi 1051 


1102 


1153 1204 1254 18(5 1356 


14;;7 61 


4 


1458 


15C9 1560 


1610 


1661 1712 1763 1814 1865 


1915 51 


5 


1966 


20171 2068 


2118 


2169: 2220 2271: 2822 2372 


2423 61 


€ 


2474 


2524 2575 


2626 


2677J 2727, 2778' 2829 2879 


2980 61 


7 


2981 


303l! 3082 8183 


8183, 8284 8285 8885 8886 


3437 51 


8 


3487 


3588 8589, 8689 


8(90 8740 3791, 8841 8892 


3948 61 


9 


3993 


4044. 4094! 4145 


4195' 4246 4296 4847 4897 


4448 51 


860 934498 


934549'934599'984650'9347C0'9S4751 934801 934852 984902 


984953' 60 


1 


6003 


5054 51041 5154 52(5, 5255 5306 5356 54C6 


5457 60 


2 


5507 


5558 5608 5658! 5709 6759 58(9. 5860 591(' 


5960 60 


3 


6011 


6061 611l' 6162 6212 6262 6818 6863 6418 


6463 60 


4 


6514 


6564 6614 6665 6715 6765 6815 6865 C916 


6966 60 


6 


7016 


7066 7117: 7167 


72171 7267, 7317 7867 7418 


7468 50 


6 


7518 


7568 7618 7668 


7718' 7769 7819 7809 7919 


7969 60 


T 


8019 8069 8119 8169 


8219 82(9 8320 8370 8420 


8470 60 


8 


8520 8570 S620| 8670 


8720: 8770 8820^ 8870 8920 


8970 60 


9 


9020 9070 9120: 9170 


9220 9270 9320' 9309 9419 


9469 60 


870 '939519989569 989619 9896e9'98971 9 9897e9 989819 989869 989918 


989968 60 


1 940018 940068 940118 940168 940218 £40267 94(317 94C307 940417 


940467 50 


2 


0516, C566 0616, 06661 0716; 0765 0815 C865 €915, 


0964 60 


3 


1014 1064 1114 1163 


1213 12f3 1313 1362 1412| 


1462 60 


4 


1511 1561 1611 


1660 


1710 


1760' 18(9 1859 1909 


1958 50 


5 


2008' 2058 2107 


2157 


2207 


2256 23061 2355 24C5 


2455 50 


6 


2504! 2554 2603 


2653 


2702 


2752 2801 2851 2901 


2950 60 


7 


3000 3049 3099 


3148 


3198 3247 3297; 3346 3896| 


3446 49 


8 


3495 3544 3593 


3643 


3692 


3742 37911 8841 3890 


3989 49 


9 
No. 


3989 4038 4088 


4137 


4186 


4236; 4285 4335 4384 


4483 40 


12 3 1 


4 


5 i 6 7 18 


9 i Diflf. 



Logarithms of IS" umbers. 



497 



No. 0|1|2|3|4|5|6|7|8|9 | Diff. 


880 '944433 944532 9445S1 944631 944630 944729 944779 94482S 944877 9449271 49 


1 


4976 5025 5074 5124 5173 5222 6272 


i 5321! 5370 


1 5419 49 1 


2 


5469 5518 5567 5616 5665 5715 5764 


5813; 5362 


5912 


49 


3 


5961 6010 6i)59 6108 61571 62o7| 6256 


6305' 6354 


' 6403; 


49 


4 


6452 6501 6551 6600 6649 6698 6747 


6796 6345 


; 6894; 


49 


5 


6943 6992 7041 7090 7140 7139 7238 


7287 7338 


7385; 


49 


6 


7434' 7483' 7532 7581 7630| 7679: 7728 


7777 7826 


i 7875; 


49 


T 


7924 7973' 8022 S070 8119 8163; 8217 


8266 8316 


83(>4 




49 


8 


8413 8462' 8511 8560| 86i,9 8657! 8706 


8765 8804 


8863 




49 


9 


8902; 8951' 8999 9048 9o97 9146 9195 


9244' 9292 


9341 




49 


890 


949390 949439 949433 949536 949535 949G34 949683949731 949730 949329 


49 


1 


9873 9926 9975 950024 950073 950121 950170 950219 950267 95U316 


49 


2 '950365 95,3414 950462 0511 0560 


1 06i.;8 0657 


0706; 0754 


0803 


49 


3 


0851 0900 0949 0997^ 1046 


1095' 1143 


1192' 1240 


1289 


49 


4 


1338 1386 1435 1483 1532 


15S0! 1629 


1677 


1726 


1775 


49 


5 


1823 1872 1920 1969' 20l7i 2o66; 2114 
2308 2356 2405 2453 2502! 2550; 2599 


2163 


2211 


2260 


48 


6 


2647 


2696 


2744 


43 


7 


2792 2841 ! 2839 2938 2936 


3034' 3083 


3181 


3180 


3228 


48 


8 


3276 3325' 3373 342l| 3470 


3518' 8566 


3615 


3663 


3711 


48 


9 


3760^ 3808; 3856 3905 3933 


4001 4049 


-4098 4146 


4194 


48 


930 


954243 954291 954339 954337 954435'954434'954532 


954530 954628 954677 


48 


1 


4725 4773 4821 4869 4918 


4986 5014 


6062 5110 


5158 


48 


2 


52)7 5255' 5303 5351 i 5399 


5447 6495 


5543 


5592 


5640 


48 


3 


5683 5736; 5734 5332 5830 


6923 6976 


6024 


6072 


6120 


48 


4 


6168 6216 6265 6313 6361 


6409 6457 


6506 


6553 


6601 


48 


5 


6649 6697 i 6745 6793 6840 


6838; 6936 


6984 


7032 


7080 


48 


6 


7123 7176' 7224 7272' 7320 


7368 7416 


7464 


7612 


7559 


48 


7 


76071 7655' 7703 7751 7799 


78471 7894 


7942 


7990 


8088 


48 


8 


8386 8134' 8181' 8229, 8277 


8325' 8373 


8421 


8468 


8516 


48 


9 


8564 8612 8659 87o7 8755 


8803I 8850 


8893 


8946 


8994 


48 


910 


959341959039 959137 959135 939232 95923o'95932S 


959375 959423 959471 


48 


1 


9518 9566' 96141 9661 9709! 9757| 9304! 9852! 99001 9947 


48 


2 


9995 960042 900090;960138 960185 960233 96U230 960328 960376960423 


48 


3 


980471 0513i 0566i 0613. 0661 


0709, 0758 


0804 


0861 


0899 


48 


4 


0946 


0994' 1041 1089 1136 


1184' 1231 


1279 


1326 


1374 


48 


5 


1421 


1469. 1516, 1563 


1611 


1658 1706 


1753 


1801 


1848 


47 


6 


1895 


1943 1993! 2033 


2085 


2132 2180 


2227 


2276 


2322 


47 


7 


2369 


24171 2464i 2511 


2559 


2606 2653 


2701 


2748 


2795 


47 


8 


2343 


2390 29371 2985 


3032 


3079 3126 


3174 


8221 


8268 


47 


9 


3316 


3363 3410j 3457 3504 


3552 8699 


3646 


8693 


3741 


47 


920 


963788 983335 933832'963929 963977'964024 96407l'964118'964165'964212 


47 


1 


4263 


4307; 4354; 4401] 4448 


4495 4542 


4590 


4637 


4684 


47 


2 


4731 


4778 4825 4372, 4919 


4966 5013 


5061 


5108 


6155 


47 


3 


6202 


5249 5298; 5343 5390 


5437 6484 


6531 


6578 


5625 


47 


4 


5672 


5719 5766 5813 


5860 


5907 5954 


6001 


6048 


6095 


47 


6 


6142 


6189 62361 6233 


63291 6376 6423 


6470 


6517 


6564 


47 


6 


6611 


6658 67051 6752 


6799! 6845; 6392 


6939 


6986 


7033 


47 


7 


7080 


7127 71731 7220 


7267; 7314' 7361 


7408 


7464 


7501 


47 


8 


7548 7595 7642 7683 


77351 7782 7829 


7875 


7922 


7969 


47 


9 


8016 8062 8109 8156 


8203' 8249, 8296 


8343 


8390 


8486 


47 


930 


963433 983530 933576 963823 963670'968716 968763 


9G8810'963S56'9G8908 


47 


1 


8950, 8998 9343 90901 9136 9183 9229 


9276, 9328i 9369 


47 


2 


9416, 9463 9509 9556' 9802 9649 9695 


9742' 9789' 9335 


47 


8 


9382 9928 9975 970021 970068 970114 970161 


970207 970264 970300 


47 


4 


970347,970393 970440 


0486 1 0533, 0579 0626 


0672 


1 0719 


0766 


46 


5 


0812 


0858 0904 


0951 


0997 1044 1090 


1137 


1 1183 


1229 


46 


6 


1276 


1322 1369 


1416 


1461 i 1508 1554 


1601 


1647 


1693 


46 


7 


1740 


1736 1832 


1879 


1925 19711 2018 


2064i 2110 


2157 


46 


8 


2208 


2249 2295 


2342 


2388 2434' 2481 


2527 


2573 


2619 


46 


9 

No. 


2666 



2712 2758 
1 1 2 


2804 
" 3 


2851'; 2897| 2943 


2989 


8086 


8082 
9 


46 


4 


1 5 1 6 


7 " 


8 


DiS 



40 s' 



498 



Logarithms of Numbers. 



No. 







iDifF. 



940 
1 
2 
3 
4 
6 
6 
7 
8 
9 

950 
1 



980 
1 



973128 973174 973220'973266 973313 973359 973405 973451 973497 973543 46 



3590 
4051 
4612 
4972 
6432 
6891 
6350 
6808 
7266 



3636 3682 

4097, 4143 

4558' 4604 

5018 5064 

54781 5524 

5937i 59S3 

6396' 6442 

6854! 6900 

7312' 78£8 



3728 3774, 3820, 3866 3913; 3959 



4189 4235 4281 i 43271 4374 

465}; 4e9o 4742: 4788 4834 

5110 5156' 5202 5248 5294 

5570 5616' 5662; 5707| 5753 

6029 6075! 6121 6167' 6212 

6488 6583| 6579; 6625 6671 

6946 6992 70371 7(1^3 7129 

7403 7419 7495 7541 7586 



977724 977769 077815 



L-, 



4420 
4880 
5340 
5799 
6258 
6717 
7175 
7632 



4005 46 

4466 46 

4926 46 

5386 46 

5845 46 

6304 46 

6763 46 

7220 46 

7678 46 



970 
1 
2 
3 
4 
5 



980 
1 
2 



8181 
8637 
9093 
9548 
980003 
0458 
C912 
1366 
1819 

982271 
2723 
3175 
3626 
4077 
4527 
49' 
5426 
5875 
6324 



8226 8272 

8683; 8728 

9138 9184 

9594 9639 
980049 98O094 

0503; 0549 

0957 1003 

14111 1456 

1864, 1909 

982316982862 

2769, 2814 

32201 3265 

3671 ! 3716 

4122! 4167 

4572 4617 



5022 
5471 
5920 



9 

990 



986772 936817 

72191 7264 

7666: 7711 

8113; 8157 

8559: 8604 

9005 j 9049 

94501 9494 

9895| 9939 

990339 9903S3 

0783, 0827 

991226 991270 

1669 1713 

21111 2156 

2554 2598 

2995 3039 

3486 348i 

8877 3921 

4817 4361 

4757 4801 

5196 5240 



5067 
5516 
5965 
6413 

986861 
7309 
7756 
8202 
8648 
9094 
9589 
9988 

990428 
0871 

991315 
1758 

2200 
2642 
8088 
8524 
3965 
4405 



977861 977906 977952 97799S 978043 978089 978135 46 



8317 8363 84(i9, 8454 

8774 8819' 8865 8911 

9280 9275 982 1| 9866 

9685 97301 9776^ 9821 



8500 8546 
8956 9002 
9412 9457 
9867 9912 
980140 980185 980281 9S0276 98U822 980367 



85911 46 

90471 46 

9503 46 

9958 46 



0594 


0640 


0685 


0730 


1048 


1093 


1139 


1184 


15011 


1547 


1592 


1637 


1954 


2000 


2045 


2C90 



07761 0821 

1229 1275 

1683 1728 

2135 2181 

982407 982452 982497 9S2543 9S25S8 982638 
2859 29 J4' 2949 1 2994 
3310 8856' 34011 3446 

8852 

4302. 4847 



8762 


3807 


4212 


4257 


4662 


4707 


5112 


5157 


5561 


5606 


6010 


6055 


6458 


6503 



4752, 4797 

5202; 5247 

56511 5696 

6100 (;i44 

6548 C59^ 



986936 9S6951 985996 987040 



8040 


3085 


3491 


3536 


8942 


8987 


4392 


4437 


4842 


4887 


5292 


5337 


5741 


5786 


6189 


6234 


6637 


6682 


987085 


987180 



980412 
0867 
1820 
1773 
2226 

982678 
3130 
3^81 
4(;'32 



45 
45 
45 
45 
45 

45 
45 
45 
45 



4482; 45 
4932 45 



5882 
5830 
6279 
6727 



7353 


7398 


7443 


7488 


7800 


7845 


7890 


7984 


8247 


8291 


8836 


8881 


8698 


8737 


8782 


8826 


9138 


9183 


9227 


9272 


9588 


9628 


9672 


9717 



75321 7577 

79791 8024 

84251 8470 

8871 8916 

9316 9361 



45 
45 

45 

45 



45 
45 
45 



990028 990072 990117 990161 990206 990250 
0472, 05161 05611 0605! 06501 0694 
C916 0960 1004| 1049| 1093| 1187 

991359 991403 991448 991492 991536 991580 



987175 
7622 
8068 

8514 45 

8960| 45 

9405 45 

9850| 44 

9902941 44 

0738, 44 

1182| 44 

991625, 44 



18021 1846; 1890| 1935j 1979 

2244 2288 2338 2877| 2421 

2686! 2730 2774 2819 2863 

8127| 8172 8216. 82601 3304 

8568: 3618; 3657 37011 3745 

4009 4053| 4097 4141 1 4185 



4449, 4493 1 4587 
4845| 4889 4933 4977 
5284 5328 5872 5116 



45811 4625 
50211 5065 
54601 5504 



2028 
2465 
2907 



2067, 44 

2509; 44 

29511 44 

8348' 8392 44 

8789! 3833, 44 

4229 4273, 44 

4669, 4713; 44 

5108, 5152 44 

55471 5591 44 



995635 995679!995723 995767 995811 995854 995898 995942 995986 996080 44 



4 
5 
6 
T 
8 
9 

No. 



6074 61171 6161 

6512 6555! 6599 

6949 6993 

7886 7430 

7823 7867 

8259 8308 

8695 8789 

9131 9174 

9565 9609 

i 1 



7037 
7474 
7910 
8347 
8782 
9218 
9652 



6215 6249 6293, 6387 6880, 6424, 6468 44 

6648^ 66871 678ll 6774 6818 6862 6906 44 

7080| 7124' 7168 7212! 72551 7299 7343 44 

75171 7561 j 7605 7648 76921 7736! 7779 44 

7954' 7998 8041^ 8085' 8129 8172 8216 44 

8890' 8434' 8477, 85211 8564 8608 8652 44 



8826, 8869 8913 8956 9000 

9261 9305! 9848 9892 9435 

9696 9789 9783, 9826, 9870 

"T'l"iri~5"' 



9043| 9087 44 
9479, 9522 44 
9918 9957 43 

9 iDiff. 



Lengths of Circular Arcs. 



499 





Table of the Lengths of Circular ^ 


4 res, radius 


being 


1 

U7lit}/. 


Seo 


\ 

;Lenffthof arc 


1 Mm. 


Length of arc. 


! Dep. 


;Lengthof arc. 


Deg. 


jLene!hofair. 


1 


0-0000048 


1 


0-0002909 


1 


i 0-0174533 


61 


1-0646509 


2 


•0000097 


2 


•0005818 


2 


•0349006 


62 


•0821042 


. 3 


•0000145 


3 


•0008727 


8 


! ^0523599 


63 


-0995575 


4 


•0000194 


4 


-0011636 


1 4 


i ^0698132 


64 


•1170108 


5 


•0000242 


5 


1 -0014544 


1 5 


•0872665 


65 


•1344641 


6 


•0000291 


^ 


! -0017453 


6 


1 ^1047198 


66 


•1519174 


7 


•0000339 


7 


•0020362 


7 


1 -1221730 


67 


•1693707 


8 


•0000388 


8 


•0023271 


8 


; •1396263 


68 


•1868240 


9 


•0000436 


9 


•0026181 


9 


' •1570796 


69 


•2042773 


10 


•0000485 


10 


•0029089 


10 


•1745329 


70 


•2217305 


11 


•0000533 


11 


•0031998 


11 


•1919862 


71 


•2391839 


12 


•0000582 


12 


•0034907 


12 


•2094395 


72 


•2566372 


13 


•0000630 


13 


•0037816 


13 


•2268928 


73 


•2740905 i 


14 


•0000679 


14 


•0040725 


14 


•2443461 


74 


•2915438 i 


15 


•0000727 


15 


•0043634 


15 


•2617994 


75 


•3089970 1 


16 


•0000776 


16 


•0046543 


16 


•279^527 


76 


•3264502 


17 


•0000824 


17 


•0049452 


17 


•2967060 


77 


•3439034 


IS 


•0000873 


18 


•0052361 


18 


•3141593 


78 


3613567 


19 


•0000921 


19 


•0055270 


i 19 


•8316136 


79 


•3788100 


2} 


•0000970 


1 20 


•0058178 


! 20 


•3490659 


80 


•3962634 ! 


21 


•0001018 


! 21 


•0061087 


21 


•8665192 


81 


•4137167 \ 


22 


•0001067 


i 22 


•0063996 


22 


•8839725 


82 


•4311700 1 


23 


•0001115 


\ 23 


•0066935 


23 


•4014258 


83 


•4486233 ; 


24 


•0001164 


1 24 


•0069814 


24 


•4188791 


84 


•4660766 i 


25 


•0001212 


1 25 


•0072723 


25 


•4363324 


85 


•4835299 ! 


26 


•0001261 


1 26 


•0075632 


26 


•4537857 


86 


•5009832 i 


27 


•0001309 


27 


•0078540 


1 27 


•4712390 


87 


•5184365 i 


23 


•0001358 


i 28 


•0081449 


28 


•4886923 


88 


•5358898 ; 


29 


•0001406 


j 29 


•0084358 


29 


•6061456 


89 


•5533431 • 


33 


•0001454 


1 30 


•0087266 


30 


•52359S8 


90 


•5707963 ! 


31 


•0001502 


31 


'0093175 


31 


•5410521 


91 


•5882496 i 


32 


•0001551 


1 32 


•0093084 


32 


•5585054 


92 


•6057029 i 


33 


•0001599 


! 33 


•0095993 


33 


•5759587 


93 


•6231562 1 


34 


•0001648 


i 34 


•0098902 


34 


•5934120 


94 


•6406095 1 


35 


•0001696 


35 


•0101811 


35 


•6108653 


95 


•6580628 i 


36 


•0001745 


36 


•0104720 


36 


•6283186 


96 


•6755561 i 


37 


•0001793 


37 


•0107629 


37 


•6457719 


97 


•6929694 ; 


38 


•0001842 


38 


•0110538 


38 


•6632252 


98 


•7104227 


39 


•0001890 


39 


•0113447 


39 


-6S06785 


99 


•7278760 


40 


•0001939 


40 


•0116355 


40 


•6981317 


100 


•7453293 , 


41 


•0001987 


41 


•0119264 


41 


•7155850 


1 


•7627826 1 


42 


•0002036 


42 


•0122173 1 


42 


•7330383 


2 


•7S02359 i 


43 


•0002084 


43 


•0125082 


43 


•7504916 


3 


•7976892 


44 


•0002133 


44 


•0127991 


44 


•7679449 


4 


•8154125 ! 


45 


•0002181 


45 


•0130900 


45 


•78539S2 


5- 


•8325958 1 


46 


•0002230 


46 


•0133809 


46 


•8028515 


6 


•8500491 


47 


•0002278 


47 


•0136718 


47 


-8203048 


7 


•8675024 


48 


•0002827 


48 


•0139627 


48 


•8377581 


8 


•8849.557 


49 


•0002375 


49 


•0142536 


49 


•8552113 


9 


•9024090 


50 


•0002 124 


50 


•0145444 


50 


•8726646 


110 


•9198622 


51 


•0002472 


51 


•0148353 


51 


•8901179 


11 


•9373155 


52 


•0002521 


52 


•0151262 


52 


•9075712 


12 


-9547688 


53 


•0002569 


53 


•0154171 


53 


•9250245 


13 


•972'2221 


54 


•0002618 


54 


•0157080 


54 


•9424778 


14 


•9896754 


55 


•0002686 


55 


•0159989 


55 


-9599311 


15 


2-0071287 


56 


•0002715 


56 


•0162898 


56 


•9773344 


16 


-024,^820 


57 


•0002763 


57 


•0165807 


57 


•9948377 


17 


•0420353 


58 


•0002812 


58 


•0168716 


58 


1-0122910 


18 


•0504886 


59 


'0002860 1 


59 


•0171625 


59 


•029744:3 


19 


•0769419 


60 


•0002909 


60 


•0174533 


60 


•0471976 


120 


•0943961 



500 




Le> 


GTHs OF Circular Ar.cs. 






1 


Table of the Lengths of Circular A 


rcs^ radms being unity. 


Dcff. 


Ler.sfliofarc 


i Deg. 
' ISl 


Length cf fxic, I 
3-15904dO ! 


De? 


Length of arc. 


DC. 


Length of arc. 


121 


2-111S4S4 1 


241 


4-2062485 


301 


5-2534411 


2 


•1293017 ! 


: 2 


•1764993 i 


2 


•2236968 


2 


•2708944 


3 


•1407550 1 


3 


•1939526 i 


8 


•2411501 


3 


•2833477 


4 


•1642083 j 


4 


•2114059 1 


4 


•2586034 


4 


•3C53010 


5 


•1S16616 1 


5 


•2288592 1 


5 


•2760567 


5 


•3232542 


G 


•1991149 


6 


•2463125 1 


6 


•2935100 


6 


•8407075 


T 


•21656S2 


7 


•2637658 1 


7 


•8109333 


7 


•3581608 


8 


•2340215 


8 


•2812191 ! 


8 


•8234166 


8 


•3756141 


9 


•2514743 j 


9 


•29367-24 | 


9 


•3458699 


9 


•3930674 


130 


•2689280 


190 


•3161256 i 


250 


•3633231 


310 


•4105207 


1 


•2863813 ! 


1 


•3335789 


1 


•3807764 


1 


•4279740 


2 


•3033346 i 


2 


•3510322 : 


2 


•3932297 


! 2 


•4454273 


3 


•3-212879 1 


3 


•3684855 i 


8 


•4156830 


3 


•4628806 


4 


•3387412 j 
•3561945 ! 


4 


•3859388 ^ 


4 


•4381363 


4 


•4808339 


5 


5 


•4033921 ! 


5 


•4505896 


5 


•4977872 


6 


•3736478 i 


6 


•4208454 i 


6 


•4680429 


6 


•5152405 


T 


•3911011 I 


7 


•4382987 i 


7 


•4854962 


7 


•5326938 


8 


•4085544 ! 


8 


•4557520 i 


8 


•5029405 


! 8 


•5501471 


9 


•4260077 ! 


9 


•4732G53 i 


9 


•5204028 


i ^ 


•5676004 


140 


•4434609 1 


200 


•4906585 i 


260 


•5378560 


320 


•5850586 


1 


•4609143 i 


i 1 


•5081118 ! 


1 


•5553C93 


1 


•6025069 


2 


•4783676 ; 


2 


•5255651 1 


2 


•5727626 


2 


•6199202 


3 


•49582U9 1 


3 


•5430184 ; 


3 


•5902160 


3 


•6374135 


4 


•5132742 


4 


•5604717 i 


4 


•6076693 


4 


•6548668 


5 


•5307274 ; 


5 


•5779250 : 


5 


•6251225 


5 


•6723201 


6 


•5481807 . 


6 


•5953783 i 


G 


•6425758 


6 


•6897744 


T 


•5656340 1 


7 


•61-28316 ' 


7 


-6600291 


7 


•7072267 


8 


•5830873 1 


8 


•6312S49 j 


8 


-6774824 


8 


•7246800 


9 


•6005406 


9 


•6477332 | 


9 


•6949357 


9 


•7421332 


150 


•8179939 


1 210 


•6651914 i 


270 


-7123890 


1 330 


•7595865 


1 


•6354472 


1 


•6826447 i 


1 


-7298423 


1 1 


•7770898 


2 


•6529005 ; 


2 


•7000930 1 


2 


•7472956 


2 


•7944931 


3 


•6703538 . 


3 


•7175513 i 


3 


•7647489 


1 3 


•8119464 


4 


•6878071 : 


4 


•7350046 ; 


4 


•7822022 


4 


•8298997 


5 


•7052604 1 


5 


•7524579 \ 


5 


-7996554 


5 


•8468530 


6 


•7227137 , 


6 


•7699112 \ 


6 


•8171087 


6 


•8643063 


T 


•7401670 ! 


7 


•7373645 ' 


7 


•8345620 


1 7 


•8817596 


8 


•7576203 


i 8 


•8043178 ; 


8 


•8520153 


i 8 


•8992129 


9 


•7750736 : 


' 9 


•8222711 i 


9 


•8694686 


9 


•9166661 


160 


•7925268 , 


. 220 


•8397243 i 


280 


•8869219 


340 


•9341194 


1 


•8099801 ^ 


1 1 


•8571776 i 


1 


•9043752 


1 


•9515727 


2 


•8274384 ; 


2 


•8746309 ; 


2 


•9218285 


2 


•9690260 


3 


•8448867 i 


': 3 


•8920842 : 


3 


-9392818 


1 3 


•9864793 


4 


•8623400 ; 


! 4 


•9095375 i 


4 


•9567351 


' 4 ■ 


6^0039326 


5 


•8797933 \ 


1 5 


•9269908 ' 


5 


•9741883 


; 5 


•0213859 


6 


•8972466 ■ 


! 6 


•9444441 : 


6 


•9916416 


i 6 


•0388392 


7 


•9146999 


! T 


•9618974 


7 


5-0090949 


7 


•0562955 


8 


•9321532 


! 8 


•9793507 1 


8 


•0264582 


, 8 


•0737458 


9 


•9496065 


: 9 


•9968040 ; 


9 


'0440015 


9 


•0911990 


170 


•9670597 


230 


4-0142572 ' 


290 


•0614548 


350 


•1086523 


1 


•9845130 


■ 1 


•0317106 1 


1 


•0789081 


1 


-1261056 


2 


3^0019667 


: 2 


-0491639 ' 


2 


•0963614 


2 


•1485589 


3 


•0194196 


' 3 


•0666172 ! 


3 


•1138147 


3 


1610122 


4 


•0368729 


' 4 


•0840705 i 


4 


•1312680 


4 


•1784655 


5 


•0543262 


5 


■1015237 i 


5 


•1487213 


5 


•1959188 


6 


•0707795 


6 


•1189770 


6 


•1661746 


6 


•2183721 


7 


•0892328 


7 


•1364303 ! 


7 


•1836279 


7 


•2308254 


8 


•1066861 


; 8 


•1538836 ' 


8 


•2010812 


8 


•2482787 


9 


•1241394 


9 


•1713369 


9 


•2185345 


9 


•2657320 


180 


•1415927 


I 240 

1 


1887902 1 

i 


800 


•2359878 


860 


•2831858 



1 



Long Chords. 



601 



EXPLANATION OF THE USES AND APPLICATIONS OF 

THE TABLE OF LONG CHORDS, 

PROBLEM. 

Required to jind the distances or abscissas on the chord from which, if 

ordinates or perpendiculars he drawn, they will pass through the 

station points on the curve. 

Example. — Let the given curve be 1000 ft. long of 5° curvature, 
or 1146 ft radius. 

For the first station from the beginning we have 

chord for 1000 ft. — chord for 800 ft. 

r:r 1st distance, 



chord 800 — chord 600 



= 2nd distance, etc. 



Then by table we have, 

968-87— 784-10 



Intermediate 
Distance. 
-= 92-885 



784-10 — 593-36 



693-36 — 398-10 



398-10- 198-81 



1 98 81 — 0000 

_ 



: 95-370 
- = 97-630 
: 99-645 
= 99-405 



Distance. 

1st dist. — 

2nd " == 

3rd " = 

4th " = 

5th " = 



484-435 =: half length 



Total. 
92-385 

187-755 

285-385 

885-030 

484-435 

968-87 



Thus for any given station we take from the length of the whole 

chord the length of a chord of twice as many stations less than the 

one under consideration; that is, 1st station from beginning 2 less; 

2 from beginning, 4 less, etc., and take half the difference. 

i If the chord had been for 900 ft. of curve, we should have, 

877-32 — 689-39 ^^ ^ , ,. 

— rr 93-965 = 1st distance. 



2 
689-39 — 496-20 



496-20-299-24 



299-24 — 100 



=r 96-595 - 2nd 
= 98-480 = 3rd 
= 99-620 = 4th 



388-660 
Add 50 — 

= 877-32 , ,^, , .. , , 

438-66 — ^ — = "^" length of chord. 



502 Middle Ordinates. 



! In like manner we may find the ordinates connecting these 
; abscissas with tlieir points on the curve. 

j Let the length of chord and radius be as already given. Then 
: we have, 

I Mid. ordmate 1000 ft. — mid. ordinate 800 ft. curve = ordi- 

nate at 1st station. 
I Mid. ordinate 1000 ft. — mid. ordinate 600 ft = ordinate at 

; 2nd station. 
i 

For this purpose we have calculated a table of middle ordinates 
corresponding to that of long chords. From this we have, 

107 -39 — 69-1.3 = 88-62 = 1st ordinate. 

107-39 — .39-06= 68'33 = 2nd 

107-39 — 17*41 == 89-98 == 3rd " 

107-39 — 4-36 = 103-03 = 4th 

107-39 — 0-00 = 107-39 == 5th or middle ordinate. 

Were the chord for 900 ft. of curve we should have by tables, 

87-25 — 53-05 = 3420 = 1st ordinate. 
87-25 — 27-17 := 60-08 = 2nd 
87-25— 9-81 =:= 77-44 = 3rd 
87-25— 1-09 = 86-16=: 4th 
87-25— 0-00 = 87-25 = middle" 

This will sufficiently demonstrate how the ordinates can be 
obtained for any other length of chord or curve. The same principle 
obtains in regard to any other rate of curvature. After passing the 
middle ordinate, their lengths will be repeated inversely ; as will 
also be the intermediate lengths of abscissas. Then from end of 
first abscissa erect first ordinate, and so on in regular rotation. 









Middle Ordinatea. 








503 




TABLE 


Of Middle Ordinates from Chords subtending Curves of from 


100 t^ low feet in length; calculated to every 15' of Curvature 


from 15' to 8°. Rad'ucs 


of 1° being 5730 feet. 


LENGTHS OF ARCS. 


100 


200 j 300 


400 


500 


600 


700 


800 


900 


1000 


MIDDLE OKDINATES. 


Curvature. 


0-06 


0-22 


0-49 


0-87 


1-86 


1-96 


2-67 


3-49 


442 


5-45 


0° 15' 


30 


Oil 


0-44 


0-98 


1-75 


2-73 


3-93 


5-34 


6-98 


8-83 10-90 


45 


0-16 


0-65 


1-47 


2-62 


4-09 


5-89 


8-01 


10-47 


18-25 16-85 


1" 00' 


0-22 


0-87 


1-96 


3-49 


5-45 


7-85^ 


10-69 


13-96 


17-67 


21-80 


15 


0-27 


1-09 


2-45 


4-36 


6-82 


9-81 


13-36 


17-44 


22-07 


27-24 


30 


033 


1-31 


2-94 


5-23 


8-18 


11-77 


16-08 


20-98 


26-48 


82-68 


45 


0-38 


1-53 


3-43 


6-11 


9-54 


13-73 


18-70 


24-41 


30-88 


8811 


2" 00' 


0-44 


1-75 


3-92 


6-98 


10-90 


15-68 


21-35 


27-88 


85-27 


43-52 


15 


0-49 


1-96 


441 


7-85 


12-26 


17-64 


24-02 


81-85 


89-66 


48-93 


30 


0-55 


2-18 


4-91 


8-72 


13-62 


19-60 


26-68 


84-82 


44-04 


54-88 


45 


0-69 


2-40 


5-40 


9-59 


14-98 


21-56 


29-83 


88-29 


48-41 


59-71 


3° 00' 


0-65 


2-62 


589 


10-46 


16-34 


23-52 


31-98 


41-74 


52-78 


65-08 


15 


0-71 


2-84 


6-38 


11-33 


17-70 


25-47 


34-63 


45-19 


57-13 


70-44 


30 


0-76 


3-05 


6-87 


12-20 


19-06 


27-42 


87-28 


48-68 


61-47 


75-78 


45 


0-82 


3-27 


7-3(> 


13-07 


20-41 


29-30 


89-92 


52-07 


65-80 


8110 


4° 00' 


0-87 


3-49 


7-85 


18 94 


21-77 


81-81 


42-56 


55-50 


70-12 


86-40 


15 


0-93 


3-71 


8-34 


1481 


2312 


3825 


45-19 


58-92 


74-48 


91-68 


30 


0-98 


3-93 


8-82 


15-68 


24-47 


35-19 


47-82 


62-84 


78-72 


96-94 


45 


1-04 


4-14 


9-32 


16-55 


25-82 


37-13 


50-44 


65-74 


82-99 


1(218 


b" 00' 


1-09 


4-36 


9-81 


17-41 


27-17 


89-06 


53-05 


69-13 


87 25 


107-39 


15 


1-15 


4-58 


10-30 


18-28 


28-52 


4(1-99 


65-67 


72-51 


90-5(1 


112-58 


30 ! 


1-20 


4-80 


10-79 


1915 


29-87 


42-92 


58-27 


75-88 


95 78 


117-75 


■ 45 i 


1-25 


5-01 


11-27 


20-01 


31-21 


44-84 


60-86 


79-25 


99-94 


122-89 


6' 00' 


1-31 


5-23 


11-76 


20-88 


32-55 


46-76 


63-45 


82-60 


104-18 


127-99 


15 


1-36 


5-45 


12-25 


21-74 


33-89 


48-67 


66-04 


85 98 


1(8-80 


188-07 


30 


1-42 


5-67 


12-74 


22-60 


35-23 


50-59 


68-62 


89-26 


112-45 


13812 


45 


1-47 


5-89 


13-23 


23-47 


36-57 


52-50 


71-18 


92-57 


116-58 


14813 


V 00' 


1-53 


o-io 


13-71 


24-33 


37 91 


54-40 


73-74 


95-87 


120-69 


14812 


15 


1-58 


6-32 


14-20 


25-19 


39-24 


56-30 


76-80 


99-15 


124-78 ir3(»7 1 


30 


1-64 


6-54 


14-69 


26-05 


40-57 


58-19 


78-84 


102-42 


128-84157 98 


45 


1-69 


6-76 


1518 


26-91 


41-90 


60-08 


81-37 


105-68 


182-88 162-86 , 


8-^ 00' 


1-75 


6-98 


15-66 


27-77 


43-23 


61-97 


83-90 


108-92 


136-89 


167-70 ; 

1 


On the principles by vMc 


i 
h the following tables are calculated. 

i 


Let m = linear opening 


1 
of switch rail, s = angular opening of 


rail, /=: angle of frog, g = 


gauge of track. 


Let X — length of ehord 


from opening of switch I'ail to point of 


frog. Then will the anioiii 


it of curvature between the opening of 


rail where curve coninienc 


^s and point of frog = / — s ; therefore 


the instrument setting over 


the open end of switch rail with a back- 


sight on the fixed end of it, 


the instrumental deflection to the point 



504 Lengths of Turnouts. 



f — s 

of frog will be = But if the backsight be taken on a point 

z 
(say 5 inches distant) parallel with the main track, the deflection will 

f 5 f-\-S 

then be = 1- s = -• Making the value of x, radius, g — m 

will be homolosjous to the sine of ^— - — Then we have, 

Sm ( ^—Y- ): R '.: g — m:x^ — ^ ^ 

\ ^ ^ sin f f -^ s' 



m 



Example : 

Calling s=:V 1 5', / == 6° 45', g = 4-70, m — 0*42, and g—'m — 
4-28, we have sin. 4° : 72 :: 4*28 : x — 61-36ft. 

When a double opening of a switch rail for a double turnout 
occurs, we have, 

//■+ 2s\ 
sin. I ■ — 7. ] : E :: g — 2x0:^:; = distance to nearest frosr. 






The linear and angular opening of rail being the same, this 
table may be adapted to any other gauge by increasing the 
value of X as given in this table, and the length of radius of turnout 
2 per cent, for every additional inch in the gauge. This is a little 
toa much ; the correction for a 6 ft. gauge being about 30 per cent. 
Thus 100 ft. chord of turnout on this track will give 130 ft. on 6 ft. 
gauge, and 1000 ft. radius will give 1300 ft. This is for a straight 
line. When on a curve going the same way as turnout, it is suffi- 
ciently accurate for practice to add rate of curve of main track to 
that of the table ; but when going in opposite direction, subtract it ; 
thus making relative departure from main track the same as on a 
straight line. 

Example : 

Thus a 5° frog for a 4ft. S^ inch gauge gives a distance of YS'S ft. 
curvature 4° 46'. If the main track were a 4° curve and going the 
same way, distance being the same, the rate of curvature would be 
4° 46' +4° = 8° 46', radius 653 ft; but going the other way 
4° 46' — 4° = 0° 46', radius 7413 ft. 



Table for Locating Turnouts. 



505 



TABLE 1 

i 
Of distances on chord from opening of switch rail where the curve \ 
commefices, to point of frog, radius of curvature and rate per \ 
100 ft, calculated to every 15 minutes of frog angle, from 3° <o 
15°. Constant data: opening of switch rail 5 inches ='4:2 ft., 
average angular opening say 1° 15', rails being from 18 ^o 20 ft. 
long. Variable data gauges of road. 







Gauge 4ft. 8^- 


inches = 4*70 ft. 






Ani^le of 
frog. 


Distances. 


Length of 
radius. 


Rate of 

curve per 

100 ft. 


Ansrle ofj 

frog. Distances. 


Length of 
radius. 


Rate of 

curve per 

100 ft. 


8° 


115-43 


3779-3 


V 


31' 


r 


47 99 


355-0 


16" 09' 


15' 


109-02 


3023-3 


1 


50 


15' 


4^6-78 


335-3 


17 07 


30' 


103-28 


2613-2 


2 


11 


30' 


45-69 


317-6 


18 04 


45' 


98-12 


2249-0 


2 


33 


45' 


44-66 


301-3 


19 02 


4^ 


93-45 


1947-2 


2 


56i 


10° 


43-67 


286-2 


20 03 


15' 


89-21 


1704-0 


3 


22 


15' 


4272 


272-2 


21 05 


80' 


85-33 


1508-0 


3 


48 


30' 


41-80 


259-3 


22 08 


45' 


8V1S 


1339-0 


4 


17 


45' 


40-95 


247-2 


23 13 


5"^ 


78-51 


1199-8 


4 


4U 


ir 


40-11 


236-0 


24 20 


15' 


75-50 


1081-6 


5 


18 


15' 


39-36 


225-4 


25 28 


SO' 


72-70 


980-3 


5 


51 


30' 


38-55 


215-8 


20 37 


45' 


70-01 


892-9 


6 


25 


45' 


37-81 


206-6 


27 48 


6° 


67-69 


816-8 


7 


01 


12" 


37-10 


198-0 


29" — 


15' 


65-44 


715-1 


7 


39 


15 


36-41 


189-5 


30 13 


30' 


63-33 


690-4 


8 


18 


30' 


35-75 


182-4 


31 27 


45' 


61-36 


639-4 


8 


58 


45' 


35-12 


175-3 


32 43 


r 


59-50 


593-0 


9 


40 


13" 


34-51 


168-6 


34 02 


15' 


57-75 


550-8 


10 


24 


15' 


3391 


162-2 


35 23 


30' 


56-01 


514-6 


11 


09 


30' 


33-34 


1563 


36 45 


45' 


54-55 


481-1 


11 


56 


45' 


32-79 


150-6 


38 08 


8^ 


53-08 


415-8 


12 


44 


14° 


32-26 


145-3 


39 32 


15' 


51-69 


423-3 


13 


35 


15' 


31-74 


140-2 


40 58 


30' 


50-36 


398-3 


14 


25^- 


30' 


31-24 


135-4 


42 26 


45' 


49-11 


375-4 


15 


17 


45' 


30-75 


130-8 


43 56 












15" 


30-28 


126-5 


45 26 



















43 



506 



Table for Locating TuRNotfxs. 









TABLE 










Of distances on chord 
frog, radius of 


from opening of sioifch rail to 
curvature and rate per 100 ft. 


pot 


it of 








Gauge 4ft. 10 inches. 






Angle of 
tVog- 


i 
Distances. 


1 

Length of 
j rudiiis. 


i Rate of 
\ curve per ; 

100 ft. ; 


i 

'Ansrle of 
1 fi-og- 


Distances. 


Lengih of 
radius 


Ra'e a^ 

curve per 

100 ft. 










49-42 


3657 


15^ 


41 


3° 


118-89 


3892- 


1° 28' i 


9" 


15' 


112-29 


3217-0 


1 47 1 


15' 


48 18 


345 3 


16 


36 


30' 


10G-3Y 
101-06 


2709- 
2316- 


2 07 ! 
2 28i 


SO' 
45' 


47 06 
46-00 


327-1 
310 3 


17 
18 


32 


29 


4^* 


96-25 


2006- 


2 51i 


10° 


44-98 


294-7 


19 


28 


15' 


91 88 


1755- 


3 16 


15' 


44-00 


280-3 


20 


27 


30' 


87-88 


1553- 


3 411 


SO' 


43-06 


267- 


21 


28i 


45' 


84-23 


1379- 


4 09i 


45' 


42 17 


254 6 


22 


3H 


5^ 


8086 


1235- 


4 38i- 


ir 


41 31 


243- 


23 


36 


15' 


77-76 


1134- 


5 03 


15' 


40 48 


232 3 


24 


42 


30' 


74-88 


1009- 


5 4Uii 


80' 


39 70 


222-2 


25 


49 


45' 


72-21 


919- 


6 14 


45' 


38-94 


212-7 


26 


58 


6^ 


69-72 


841- 


6 49 


12° 


38-21 


203-9 


28 


09 


15' 


67-40 


772- 


7 25^ J 


15' 


37 50 


1955 


29 


21 


30' 


65-22 


712- 


8 03 


30' 


36 82 


187 8 


30 


33 


45' 


63-20 


658- 


8 42i 


45' 


36-17 


180 5 


31 


46 


r 


61-28 


610 


9 23^ 


13° 


35-54 


173 6 


33 


00 


15' 


59-48 


568- 


10 06 


15' 


34-92 


167- 


34 


18 


30' 


57-79 


530- 


10 50 


30' 


34-34 


160-9 


35 


39 


45' 


56-18 


495-5 


11 35 


45' 


33-77 


155- 


37 


00 


8° 


54-67 


464-3 


12 21 


14° 


33 22 


1496 


38 


20 


15' 


53-24 


436- 


13 09 


15' 


32-69 


144-4 


39 


44 


30' 


51-87 


410-2 


13 59 


30' 


32-17 


139-4 


41 


10 


45' 


50-58 


386 6 


14 50 


45' 


31-67 


134-7 


42 


36 








1 


15° 


31-18 


130-2 


44 


04 


\ 



Table for Locating Turnouts. 



501 



TABLE 

Of distances on chord from opening of switch rail to point of frog ^ 
radius of curvature and rate per 100 ft. 









Gauge 5 feet. 








Angle of 
frog. 


1 
Distances. 


1 Rate of 
Length of curve per 
radius. lOO it. 


Angle of 
frog, j Distances. 


Length of 
radius. 


Rate of 

curve per 

100 ft. 


3° 


123-51 


4036 


• r25i' 


9° 


51-24 


1 379 9 


15' 05' 


15' 


116-65 


3436 


1 40 


15' 


50-00 


358-7 


15 58 


30' 


110-50 


2810 


2 02 


30' 


48-88 


339-8 


16 52 


45' 


104-98 


2403 


2 23 


45' 


47-78 


322-3 


17 48 


4* 


100-00 


2080 


2 45 


10° 


46-72 


306-2 


18 44 


15' 


95-45 


1820 


3 08|- 


15' 


45 71 


291-2 


19 42 


30' 


9130 


1611 


3 33 


30' 


44-73 


277-4 


20 40 


45' 


87-50 


1430 


4 00 


45' 


43-81 


264-5 


21 40 


S'' 


84- 


1281 


4 28 


ir 


42-91 


252-5 


22 42 


15' 


8078 


1156 


4 57 


15' 


4200 


241-2 


23 46 


30' 


77-78 


1047- 


5 27 


30' 


41-24 


230-9 


24 52 


45' 


75-00 


965 


5 58 


45' 


40-45 


221-0 


26 01 


6° 


72-32 


873 


6 33^ 


12° 


39-69 


211-8 


27 10 


15' 


70-00 


802- 


7 09 


15' 


38-95 


202-7 


28 20 


30' 


67-76 


739- 


7 45 


30' 


38-25 


195-1 


29 30 


45' 


65-65 


684- 


8 23 


45' 


37-57 


187-5 


30 40 


r 


63-66 


634- 


9 02 


13° 


36-92 


180-2 


31 50 


15' 


61-78 


590- 


9 43 1 


15' 


36-28 


173-5 


33 02 


30' 


60-00 


550- 


10 25 


30' 


35 67 


167-2 


34 17 


45' 


58-36 


514- 


11 09 


45' 


35-08 


161 1 


35 85 


8^ 


56-79 


482- 


11 51 


14° 


3451 


1554 


36 55 


15' 


55-30 


452- 


12 40 


15- 


33-96 


150 


38 16 


30' 


53-88 


426- 


13 27 


30' 


33-42 


144-8 


39 38 


45' 


52-54 


401- 


14 17 


45' 


32-90 


139 9 


41 00 










15" 1 


32-39 


135-3 


42 23 



508 



Table for Locating Turnouts. 











TABLE 






Of distances on chord from, opening of stvitch rail to point 
radius of curvature and rate per 100 feet. 


<]ff^09y 


Gauge 5 feet 6 inches. 


Angle of 
frog. 


Distances. 


Length of 
radius. 


Rate of 

curve per 

100 ft. 


Angle of 
1 frog. 


Distances. 


Length of 
radius. 


Rate of 

curve per 

100 ft. 


3° 


136-'78 


4478- 


r 


17' 


•: 9° 


56-87 


420-7 


13° 39' 


15' 


129-19 


3750- 


1 


32 


15' 


55-40 


397-4 


14 27 


30' 


122-38 


3116- 


1 


50 


30' 


54-14 


376-4 


15 14 


45' 


116-27 


2664- 


2 


09 


45' 


52-92 


357-0 


16 04 


4°, 


iio-'zs 


2307- 


2 


29 


10° 


5174 


339-1 


16 55 


15' 


105-71 


2019- 


2 


50 


15' 


50-62 


322-5 


17 47 


30' 


101 11 


1786- 


3 


12 


30' 


49-54 


307-2 


18 40 


45' 


96 90 


1586- 


3 


37 


45' 


48-52 


292-9 


19 35 


5° 


9303 


1421-0 


4 


02 


11° 


47-52 


280-0 


20 30 


15' 


89-46 


1281- 


4 


28 


15 


46 52 


267-2 


21 28 


80' 


86-14 


1161- 


4 


56 


30- 


45-68 


255-7 


22 26 


45' 


83-15 


1062- 


5 


24 


45' 


44-80 


244 8 


23 26 


6° 


80-16 


967- 


5 


56 


12° 


43-96 


234-2 


24 30 


15' 


77-53 


888-8 


6 


27. 


15' 


43-14 


224-7 


25 33 


30' 


75-04 1 


819- 


7 


00 


30' 


42-36 


215-9 


26 36 


45' 


72-71 


757-6 


7 


34 


45' 


41-61 


207-7 


27 40 


1° 


70-50 


702-8 


8 


10 


13° 


40-89 


199-7 


28 46 


]5' 


68 43 


6538 


8 


46 


15' 


40-18 


192-2 


29 54 


30' 


66 47 


609 8 


9 


24 


30' 


39-50 


185-2 


31 02 


45' 


64 64 


5700 


10 


04 


45' 


38-85 


178-4 


32 11 


8' 


62 89 


534- 


10 


45 


14° 


38-22 


172-1 


33 21 


15' 


61-25 


501-6 


11 


27 


15' 


37-61 


166-1 


34 33 


30' 


59 67 


471-9 


12 


10 


30' 


37-01 


160-4 


35 47 


45' 


5819 


444-8 


12 


54i 


1 45' 


36-44 


154-9 


37 03 












15° j 31-87 


150-0 


38 18 



^ 



Table for Locating Turnolts. 



509 











TABLE 










Of difitances on chord from opetu 
radius of curvature 


ng of switch rail to point of frog ^ 
and rate per 100 /iJ. 








Gauge 


6 feet. 




1 


Angle of 
frog. 


Distances. 


Length of 
radius. 


Rate of 

curve per 
100 ft. 


Angle of 
frog. 


Distances. 


Length of 
radius. 


Rate of 
curve per 

100 ft. 


3" 


150-06 


4913-1 


V 


10' 


9' 


62-40 


4616 


12^ 


26' 


15' 


ui-Ys 


4060-3 


1 


24| 


15' 


60-81 


4359 


13 


10 


30' 


134-26 


3419-3 


1 


40i 


30' 


59 40 


412-9 


13 


55 


45' 


127-56 


2923-7 


1 


57i 


45' 


58-06 


391-7 


14 


40 


4^ 


121-50 


2531-4 


2 


16 


10° 


56-77 


3721 


15 


25 


15' 


115-97 


2215-2 


2 


35 


15' 


55-54 


353 9 


16 


12 


30' 


110-93 


19604 


2 


55i 


30' 


54-35 


337 1 


17 


00 


45' 


10631 


1740-7 


3 


17i 


45' 


53-24 


321-4 


17 


50 


5° 


102-06 


1560-0 


3 


m 


ir 


52-14 


306-8 


18 


42 


15' 


98-15 


1406-1 


4 


o^ 


15' 


51-04 


293-2 


19 


34 


30' 


94-51 


12744 


4 


30 


30' 


5012 


280-5 


20 


27 


45' 


9M4 


1160-8 


4 


56 


45' 


49 15 


2686 


21 


22 


6° 


88-00 


1061-8 


5 


24 


12° 


48-23 


257-4 


22 


18 


15' 


85-07 


975-0 


5 


53 


15' 


47-33 


2460 


23 


15 


30' 


82'33 


898-8 


6 


23 


30' 


46-47 


237-1 


24 


12 


45' 


79-77 


831-2 


6 


54 


45' 


45-66 


227-9 


25 


12 


r 


77-35 


771-0 


7 


26 


13° 


44-86 


219-2 


26 


12 


15' 


75-08 


717-3 


8 


00 


15' 


44-08 


210-9 


27 


14 


30' 


72-94 


669-0 


8 


34 


30' 


43-34 


203-2 


28 


17 


45' 


70-92 


625-4 


9 


10 


45' 


42-63 


195-8 


29 


20 


8° 


69-00 


586-0 


9 


47i 


14° 


41-94 


188-9 


30 


23 


15' 


67-20 


550-3 


10 


25i 


15' 


41-26 


182-3 


31 


28 


30' 


65-47 


517-8 11 

1 


05 


30' 


40 61 


176-0 


32 


3d 


45' 


63-84 


488-0 11 


46 


45' 


39-98 


1700 


33 


45 












15° 


39-36 


164-5 


33 


54 



4o^ 



510 Miscellaneous Notes and Examples. 



MISCELLAITEOUS NOTES AND EXAMPLES. 

Suppose a curve contain 57° 24' curvature, distance between 
centres of inner and outer track 5ft. Required difference between 
main and outside track. By table of circular arcs: 
67° gives 0-9948377 
24° " 0- 0069814 
1-OU18191 
Multiply 5 

5-0090955 

Ana. 5 ft. 

To find the length of any circular arc, multiply tabular arc of 
given number of degrees by the radius. Half of this tabular length 
gives the tabular area of a section of some number of degrees, and 
this tabular area multiplied by the square of radius, gives the 
required area of sector ; or this tabular area, multiplied by the 
difference of the squares of the two radii, gives the area of a ring. 
Thus if inner radius = 3 ft, outer = 4, thickness being 1, we have 
4^ — 3^ =r 7, which multiplied by tabular area gives area required. 
Suppose the radius of the intrados of an arch containing 134° 46' 
is 6'3 ft., the thickness of voussoirs = I'o. 

ThenS'^— 6-5'r= 21-75. 

134° gives 2-3387412 
46' " 0-0133809 
134° 46' " 2-3521221 x 21-75 = 51 '16 nearly, 

and =: 25-08 = area. 

2 

When the span and rise are given to find the curvature of arc, 

rise 

make z — ; :; — nat. tansj. 1 curvature. 

half span 

Example. — Suppose span =j: 18 ft., rise = 6 ft., then | = 0*666667 
= nat. tang. 33° 4H', and 33° 4H' x 4 = 134° 46' of curvature. 
Let it be required to find radius, we would then have, 

(i span)^ + (rise)^ ,. 9'' + 6^^ ^ .^ ,. . 

- — - — - — r--^^ r= radius. Thus :; == 9-75 = radius of arc. 

2 X rise 2x6 

Had it been a 1 2 ft. span and 4 ft. rise, radius would have been 
6*5 feet. 

Analogous to this last example, and derived from the same propo- 
sition of geometry, is an easy method of determining the distance 
across a river or ravine. 

Let the instrument be at B with a foresight upon C across river ; 
from B lay off a right angle to D. Set the instrument over D and 



Miscellaneous Notes and Examples. oil 



lay off from D C a right angle I) A meeting C B produced in A. I 
Then by similar triangles, | 

A B : B D :: B D : B C; or — ^ r= B C. Suppose that B D r:r ' 

2500 
50 ft. and A B =r 3 ft, then --— = 833 '3 ft. \ 

! 

To Triangulate round an Obstruction on a Curve. 

Example. — Suppose in running a 3° curve, I find the point for 
sta. 2645 to be occupied by a house ; I find, however, that 2644 + 75 
and 2645 + 25 are clear of the house ; also, that I have sufficient 
room for an equilateral triangle whose sides are 50 ft. each. Es- 
tablish 2644 + 75 and set the instrument over it. IS^ow suppose the \ 
last reliable point on curve to be at sta. 2640. The instrumental : 
deflection from 2640 to 2645 + 25 = 525 ft. is 7° 52^'. Set the ' 
vernier to this reading, and clamp the instriiment with a backsi_o;ht j 
on 2640, so that, when the vernier is at 0, the telescope may point 
towards 2645 + 25. Unclamp vernier, set the reading at 60°, and ; 
measure 50 ft. in line of telescope. Set instrument over this point, ■ 
and turn the interior angle = 60°, measuring 50 ft. as before. Set I 
the transit over this last point, sta. 2645 + 25, with the vernier at I 
60° so that the zero line shall coincide with the chord from 2644 + ' 
75 to 2045 + 25. Clamp the instrument with a sight on the second | 
point or vertex of triangle. Then set the vernier at 1° 52^', the ! 
instrumental deflection for 125 ft., and the telescope will point in ; 
direction of sta. 2646, from whence continue the curve, if required, i 
as before. ! 

This was an expedient applied to advantage by a former associate ' 
in making the final location of the Ohio and Mississippi R. 11., i 
Ripley County, Indiana. ! 

Similar examples and corollaries to previous propositions might | 
be added indefinitely, but this would transcend the proper limits ! 
of the work. To an adept practitioner possessing ordinary faculties | 
of generalization, it is believed the rules and formulas already given i 
will be suggestive of the means of solving most of the other problems ' 
which may occur in practice. 



^ 



TABLE FOR CURVING RAILROAD IRON. 



BY CHARLES HASLETT, Cm 



EN to tlie nearest sixteenth of an inch, the spring of a 10 feet rail being one 
at of a 20 feet rail ; the spring of a 9 feet rail being one fourth that of an 18 
When the rail is properly curved, the spring at one fourth the length from 
3f the rail, will be three-fourths that at the middle. At 5, 10, and 15 feet of a 
■ail, the spring for a 19 degree curve (301 feet radius) would be li inches, 2 



inches, and li- inches. The spring will be tested at each point by a suitable straigh 
Where the sign + occurs, curve a little more ; and where the sign — occui's, c 

little less. 

Opposite the rate of curvature and under the length of rail, will be found I 

quired spring, in inches and parts of an inch. 



LENGTH OF RAIL IN FEET. 



\r^ 



HtT 



\+ 


i 




+ + 




+ + 




iV- 




,4- 




,%- 



1*^- 



it 



'i + 

H 
1^ 



'^1 






19 


20 


-- 


i- 


- + 




lli— 




f + 


1 + 


\ + 


TT— 




fr 




tV + 


fV + 


t- 


i — 


i + 




tV— 


^ + 


F 


4+ 


U 






\ 




- + 


¥l 


s:+ 




] + 


4+ 








■'■ + 


— + 


-»~ 




w^ 


-s- + 






i + 


-f+ 


il- 






^ 


\-^-v 









1 -t-i 


..„-f 


u 



'h 



1 + 



;} + 



24 


25 


T + 


tV- 


tV 


tV + 


i __ 


\ 


i + 




fV-- 


fV + 


fV + 


i- 










4+ 




--4 


- + 


-6 


- + 


# + 


-i 


4+ 


\- + 




--* 


;l^ 




- + 


I + 


1 — 




1 + 




ItV 


1 + 


if- 


1-1 — 


't + 


;^+ 


if- 


\'f~- 


A + 


;f- 


if- 


iTTT 


IJ" 





26 


27 


tV~ 


A 


-■ + 


f^ 


f + 


1 + 








1 



r 



Ti 1 4 + 


r 


-ri, 


H 


^H-;j , . 


!^ 


1 +4"— 


ifV 


tV+^H + 


u 



-A,+ 1 T JY' _ 1 






^1 + 



lii- 

If 






li 









+I2il + 

-\-\m 



ui-i^Iii+io'" ^'A-'^iv+i^l +;4l+|2^t- 

lii+ili"_!2 +2,\+;-'| +!2,%+;2-f +J2|-t + 



-lf|-|-j2i + 



26 I 27 I 28 i 29 



INDEX, 



Accumulated Work, 66* 
. Adhesion, 176. 
Air, weight of given bulks of, 63 ; ex- 
pansion of, by heat, 231 ; properties of 
I atmospheric, 277. 

Alabaster, to harden and polish, 67. 
< Algebraic symbols, 15. 

Alloys, 286 ; recipes for various, 154-159. 
■ Alto-relievo, engraving in, 222. 
Amalgams, 89. 

Amonton's law of elastic force, 177. 
Analysis of various organic substances, 

269. 
Anchors, size of, proportioned to the 

tonnage of vessels, 272. 
Animals, work of, 55. 
Approximations, decimal, 41. 
Arches, hemispherical, how to construct, 

295. 
Architectural terms, vocabulary of, 331- 

341. 
Arcs, circular, table of the lengths of, 

499. 
Arithmetic, decimal, 24; instrumental, 

42. 
Arithmetical signs, 13. 
Artificial stones, recipes for making 

different kinds of, 862-365. 
Atmosphere, temperature and weight 

of the, at various heights, 279. 
Attraction, centre of, 175 ; capillary, 224. 
Axle grease, 283. 

Balls, cast-iron, weight of, 129. 

Beam, cast-iron, pressure it will sustain, 
146. 

Beams, equilibrium and pressure of, 59 ; 
transverse strength of, 209-211, 247; 
deflection of, 212.^ 

Bending and gluing-up, method of join- 
ing woodwork by, 328, 829. 

Binary compounds, 250-255. 

Black, for miniature pain'ers, 203. 

Blacking, recipes for, 2G2, 2C3. 

Blacklead pencils, 2u3. 

Board measure, 146. 

Boiler tubes, table of surface of, 241. 

Boilers, engine, 47 ; the, of steam- 
engines, and their proportions, 77. 



Boiling point, height of, at different 
heights of the barometer, 287. 

Bolts and nuts, sizes of, 288. 

Bookbinders' recipes, 352-358. 

Boyle's law of elastic force, 177. 

Brass, weight of superficial foot of, 144. 

Bronze powder, 159. ; 

Bronze, recipes^or various kinds of, ! 
847, 348. i 

Bronzing liquids, 108 ; for gun barrels, ; 
179 ; for tin castings, ib. ' 

Bricklaying, 297. 

Building, terms used in, 3ol-341. 

Cables, strength of, 271 ; size of, accord- 
ing to the tonnage of vessels, 272, 

Capillary attraction, 224. 

Carbon, 173. 

Carpentry, 305. 

Cast-iron, dimensions of cylindrical 
columns of, 142 ; compressio'n ot;2o9 ; 
bars, breaking weight of, 207; bars, 
transverse strength of, 211 ; beam, 
pressure it will sustain, 140 ; pillars, 
table of ultimate breaking weight in 
tons, 119-122 : pillars, strength of, 216, 
217; pipes, 148, 149; pipes, table of 
weight of, 131; plates, weiirlit of, 245; 
shafts, table of strengths of, 128, 124; 
wheel, to find the horse-power it is 
capable of transmitting, 246. 

Cement, 800, 302. 

Cements, recipes for various, 262-269. 

Centre of attraction, equilibrium, fric- 
tion, gravity, and gyration, 175, 176. 

Centre of gyration, 53*; of percussion, ib. 

Centrifugal force, 50. 

Chain rigging, scale of proofs for, 291. 

Chains and ropes, comparative scale of, 
247. 

Chalk drawings, how to render perma- 
nent, 358. 

Chords, table of, 414; uses and applica- 
tions of the table of long, 501. 

Circles, table of the diameters, circum- 
ferences, and areas of, 160-166; cir- 
cumferences and areas of, from one to 
fifty feet, 167-173. 

Cisterns, capacity of, in gallons, 274. 



i 



514 



Index. 



Clothes balls, 159. 

Coal, number of cubic feet to a ton, 2S0. 

Cocks, bore and weight of, 184, 

Cog wheels, pitch of, 68. 

Cohesion, 17(5. 

Coins, method of taking impressions 
from, 118. 

Combustion, spontaneous, to prevent. 79. 

Comi)osition, for covering buildings, 268 ; 
for walls, 3U4 ; and resolution of forces, 
312. 

Compounds, binary, 250-255. 

Compression, 176. 

Conducting powers of substances, 290. 

Copper, liquids for bronzing medals, 
figures, <fcc., of, 108; tubiog, weight of, 
134; weight of superficial foot of, 144; 
beau and feather shot, 152; etching 
fluid for, 214; rods or bolts, weight of, 
273; pipes, weight of. 274; sheet, 
weight of a square foot of, 275 ; dimen- 
sions and weight of, 138. 

Crane, to calculate the different parts of, 
as respects mechanical advantage, 58. 

Crayons, recipes for, 858. 

Crystals, diamond prismatic, for win- 
dows, 82. 

Cubes and cube roots of numbers, table 
of, 180-201. 

Cubic measure, 147. 

Cupolas, plan to take awav the lateral 
thrust of, 295. 

Cylinders, cast metal, 148. 

Dalton's law of elastic force, 177; ex- 
periment on gases, 178. 

Dampness, how to preserve walls from, 
153. 

Decay of wood, means of preventing, 
307. 

Decimal arithmetic, 24 ; approximations, 
41. 

Decomposition, vegetable, circum- 
stances favorable to, 305, 3C6, 

Diamond crystals, prismatic, for win- 
dows, 82. 

Digging, measurement of, 301, 

Distance, how to estimate, 82. 

Domes, plan to take away the lateral 
thrust 0^. 295. 

Dovetailing, process of, 324. 

Drawings, chalk and pencil, how to ren- 
der permanent, 358. 

Driving wheels, revolutions per mile of, 
86. 

Drums, velocity of, 68. 

Ductility of metals, 289, 290. 

Dyeing hats, recipe for, 256; recipes for 
compound colors in dyeing, 850, 851. 

I Earthenware, glaze for, 102. 

I Elastic force oif steam. 80 ; fluids, mecha- 

i nical laws of, 177-179. 

1 Elasticity of torsion, 148. 



Electrical machines, amalgam for, 69. 

Electrotype manipulations, solutions 
used in, ir,9. 

Enamel, various recipes for making, 848, 
849. 

Engine, horse-power of, 57; boilers, 47; 
locomotive, general proportions of, 76. 

Engraving in alto-relievo, 222. 

Engravings, transfer of, to plaster casts, 
129 ; how to take casts of, 133. 

Equilibrium, centre of, 175, of beams, 
59. 

Equivalents of sixty-two simple sub- 
stances. 248. 

Etching fluid for copper and steel, 214. -. 

Extension, 176. 

Factitious stones, recipes for making 
different kinds of, 862-365. 

Feeding properties of different vege- 
tables, 249. 

Ferrules for tool handles, solder for, ISO. 

Feet, table of solid, 245. 

Figures, copper, bronzing liquids for, 
1C8. 

Floors, the construction of, 318-320. 

Fluids, elastic, mechanical laws of, 177- 
179. 

FormulaB for running lines, locating 
side-tracks, &c., 375. 

Forces, parallelogram of, 59. 

Fresco painting, 132. 

Friction, centre of, 175. 

Fuel, power of various species of, 288. 

Furniture oil, 127; paste, 159. 

Fusing point and fusibility of metals, 
252, 284. 

Gases, illuminating, 178; properties of, 

278; pressure at which certain, are 

liquified. 257. 
Gauge-points for the engineer's and 

common slide rule, 45. 
Gay-Lussac's law of elastic force, 177. 
Geometry, practical, 16. 
Gilding, recipes for different kinds of. 

359-361. 
Glass, perfectly black hard, 82; soluble, 

222 ; recipes for making different kinds 

of 242; how to stain various colors, 

861, 362 ; paper, how to make, 304. 
Glazes, 102. 

Globes, liquid amalgam for silvering, 89. 
Glue, 263, 264. 267 ;'" recipes for various 

kinds of. 842. 
Gluing joints, process of, 323. 
Gold lustre for stone-ware, 70. 
Gradients, table of, and resistance per 

ton for each, 118. 
Gravities, specific, of simple substances, 

248; specific, of bodies, 281. 
Gravity, 47; specific, 61; centre of, 175. 
Grease', scourinsr drops for remo>ing, 

108; for axles,"'283. 



Index. 



515 



Grooving and lapping, method of join- 
ing woodwork by, 828. 

Gun barrels, bronzing liquids for, 179. 

Gunpowder, composition of different 
kinds of, 285. 

Gutta percba, solvents for, 179. 

Gyration, centre o^ 53, 176. 

Hardness of metals, ratio of the, 23. 

Hardnesses of bodies, ratios of, 289. 

Hats, recipe for dyeing, 256. 

Heat, latent, iu steam, 72; properties 
and miscellaneous effects of, 228-231 ; 
capacities of bodies for transmitting, 
235 ; power of various substances to 
transmit, 291. 

Heating power of various combustible 
substances, 82. 

Heel balls, 103. 

Horse-power, 55; of an engine, 57; of a 
steam-engine, rule for obtaining, 75; 
nominal, of low pressure engines, 83 ; 
of high pressure, 84: capable of being 
transmitted by a cast-iron wheel, to 
find the, 246; of wheels, proportionate 
strength of, 258 ; comparison of, as ap- 
plied to machinery, 286. 

Horses, traction of, 56. 

Illuminating gases, 178. 

Inches, table of solid, 245. 

Inclined plane, the, 66. 

Ink, recipes for making different kinds 
of, 865-3C9. 

Instrumental arithmetic, 42. 

Instruments, copper, bronzing liquids 
for, 108. 

Iron, to preserve from rust, 153; cast- 
iron pillars, table of ultimate breaking 
weight in tons, 119-122; cast, com- 
pression of, 209 ; cast and wrought iron 
shafts, tables of strengths of, 123-126 ; 
flat and rolled, table of weight of, 130 ; 
malleable, weight of one foot length of, 
132; sheet, weight of superficial foot 
of, 144 ; sheet and boiler plate, weight 
of a square foot of, 275; wrought, 
square, round, and flat, tables of 
weight of, 97-117 ; wrought, tensile 
and compressive strength of, 208. 

Ivory, recipes for softening, hardening, 
dyeing, marking, &c., 174, 175. 

Joiners' work, finishing of, 330, 331, 
Joints, method of gluing, 323. 
Journals of shafts, strength of, 149. 

Knot table, 259-261. 

Kyan, Mr., his method of seasoning tim- 
ber, 310-312. 

Lac, preparations of, 89. 
Land-surveying, 46. 
Latent heat in steam, 72. 



Lateral pressure, resistance to, 136. 

Laws of motion, 47. 

Lead and lead pipe, weight of, 134; 
weight of thicknesses of, 145; sheet, 
weight of a square foot of, 275. 

Lever, the, 63. 

Lime and hair, 301. 

Lines of railroad, formulae for running, 

Linseed oil, for varnishing, 87. 

Liquids, table of properties of, 283; 
bronzing, 108. 

Locomotive engines, general propor- 
tions of, 76. 

Logarithms, use of the tables of, 219- 
222 ; tables of, from 1 to 1U,000, 483. 

Lustre, gold, for stone-ware, 70. 

Lutes, fire, 266. 

Machinery, estimated power of man or 
horse as applied to, 286. 

Machines, work of, 56. 

Malleability of metals, 289, 290. 

Maltha, or Greek mastic, 308. 

Man, units of work of a, 55 ; power, com- 
parison of, as applied to machinery, 
286. 

Map colors, 143. 

Marble, plaster imitation of, 303 ; recipes 
for staining and cleaning, 349. 

Mariotte's law of elastic force, 177. 

Materials, strength of, 185; strain and 
stress of, 204; of construction, weight 
of, 234 ; conducting power of, 235. 

Masonry, different kinds of, 292, 293. 

Mastic, Greek, 803. 

Mechanical powers, the, and their appli- 
cation, 63. 

Medals, copper, bronzing liquids for, 108. 

Mensuration of superficies, 27, 44; of 
solids, 36, 46 ; of timber, 146 ; of circles, 
160. 

Metals, ratio of the hardness of, 23; 
weight of, 97 ; paste for cleaning, 146 ; 
and'other substances, weight of, 236 ; 
expansion of, by heat, 228 ; linear 
dilatation of, by heat, 230 ; fusing point 
and fusibility of, 282, 284: ductility 
and malleability of, 289, 290. 

Modelling wax, 86. 

Motion, laws of, 47. 

Mortar, 300. 

Mortice and tenon, 826^28. 

Numbers, properties of, 238, 239; table) 
of useful, 240, 241; tabic of prime 
numbers to 5000, 243-245. 

Numeration, 42. 

Nuts, sizes of, equal in strength to their 
bolts, 286. 



Oakum, weight of a cubic foot of, ISO. 
Oil, linseed, for varnishing, 87. 



616 



Index. 



Oil for furniture, 127 ; for watchmakers, 
132 ; number of gallons in a cubic foot 
of, 280. 
Organic substances, analysis of various, 
j 269. 
; Oscillation, centre of, 176. 

I 

: Paint, old, solvent for, 102. 
: Paper glass and stone, how to make, 804' 
; Papers, recipes for preparing, for various 
purposes, 342-347. 

Parallel motion, table to determine dis- 
tances of movable points in a, 223. 

Parallelogram of forces, 59. 

Partitions, timber, 322, 323. 
; Paste for cleaning metals, 146 ; for furni- 
ture, 158; common, 265; for laying 
cloth or leather on table tops, 267. 

Pencil drawings, how to render perma- 
nent, 358. 

Pencils, blacklead, 203. 

Pendulum, oscillations of, 49, 50. 

Pendulums, 249. 

Percussion, centre of, 53. 

Petrifaction of wood, 70. 

Pews composition for covering build- 
ings, 268. 

Photos-raphs, paper for, 345-347. 
■ Pile-driving, force in, 139. 

Pillars, cast-iron, table of ultimate break- 
ing weight in tons, 119-122. 

Pipe, lead, weight of, 134. 

Pipes, weight of, of various metals, 128 ; 
cast-iron, table of weight of, 131 ; cast- 
iron, 148, 149. 

Pitch of cog wheels, 68. 

Plane, the inclined, 66. 

Plaster, in imitation of marble, 303, 

Plastering, 300. 

Plastic figures, amalgam for varnishing, 
89. 

Platina for springs, 145. 

Platina-mohr, 176. 

Polishes, recipes for various, 351, 852. 

Polishing wood, recipes for, 226, 227. 

Porcelain, glaze for, 102. 

Porphyry, to cut, with iron, 146. 

Power, steam, 71. 

Power of man or horse as applied to 
machinery, 286. 

Powers, the mechanical, and their ap- 
plication, 63. 

Practical geometry, 16. 

Pressure of beams, 59 : lateral, 136 ; ver- 
tical, 141 : of steam at different tem- 
peratures, 78. 

Prime numbers, table of, to50C0, 243-245. 

Pulley, the, 65. 

Pulleys, velocity of, 68, 

Pump, common, principle of, 281. 

Pumps, quantity of water per linear foot, 
in, 276. 

Putty, old, solvent for, 102. 

Eadiating powers of substances, 290. 



Radii, table of, and their logarithms, 409. 

Eailroads, formulae for running lines, 
locating side-tracks, &c. 

Ecduction, 24. 

Iteflecting powers of various substances, 
291. 

Ptesolution of forces, 312. 

Kigging, chain, scale of proofs for, 291. 

Poofs, the construction of, 314-317. 

Ropes, strength of, 218, 271 ; comparative 
strength and weight of, 136 ; hempen, 
capacity of sustaining strain, 273; and 
chains, comparative scale of, 247. 

Rule, slide, description of, and manner 
of using, 42. 

Rust, to preserve iron from, 153. 

Safety valve, rule to find the proper! 
diameter of, 75. ' 

Scouring balls, 159. 

Scouring drops for removing grease, 103. 

Screw, the, 67. 

Screw-cutting, change wheels for, 279. 

Screws, Y-thread, number of threads to 
an inch in, 277. 

Scribing, the operation of, 829. 

Sealing-wax, recipes for making different i 
kind's of, 369, 370. 

Seasoning timber, 307-312. 

Secants, external, table of, 415. 

Seed-lac, 89. 

Shafts, cast and wrought-iron, tables of 
strengths of, 128-126; strength of 
journals of, 149 ; diameters of, 151 ; re- 
sistance of, to torsion, 150. 

Shears, tension of the guise and shear- 
leg oi; 60. 

Sheet-iron, weight of superficial foot of, 
144. 

Shell-lac, 89. 

Side-tracks, on railroads, formula for 
locating, 372. 

Siguatuiss, method of taking fac-similes 
of, 132. 

Signs, arithmetical, 13. 

Sitica, 173. 

Silver, an indelible black to write on, 
81 ; recipes for purifying and reducing, 
201, 

Silvering powder, 145. 

Sines, versed, table of, 415; natural, 457. 

Size, 173. 

Slide rule, description of, and manner 
of using, 42. 

Solder for tortoise-shell, 128 ; for ferrules 
oftool handles, 133. 

Solders, recipes for various, 235. 

Solid measure, 147. 

Solids, mensuration of, 36, 46. 

Solvents for gutta percha, 179. 

Sound, velocity of, 176. 

Specific gravity, 61 ; gravities, table of, 
62; of simple substances, 248; specific 
gravities and other properties, 281. 



Index. 



517 



Spontaneous combustion, to prevent, 79. 

Springs, platina, 145. 

Squares and square roots of numbers, 

table of, 180-201. 
Staining woods, processes for, 218, 219. 
Stains, method of removing, 179 ; vari- 
i ous, for glass, 361, 362. 
• Steam, pressure of, at different tem- 
. peratures, 78 ; elastic force of, 80 ; 
force and temperature of, in atmo- 
spheres, 81 ; pressure of, 87, 88. 

■ Steam-engine, 71 ; table of surface of 

boile? tubes of, 241 ; various problems 

i concerning, 227, 228. | 

Steam-engines, power of, 46; nominal ' 

horse-power of low pressure, 83 ; of 

high pressure, 84 ; proportions of con- : 

! densing, 85. i 

\ Steam power, 71. j 

Steel, recipe to preserve articles made j 

I of, 178; etching fluid for, 214; cast, ; 

1 composition for welding, 263. i 

! Stick-lac, 89. ■ 

I Stone, methods of joining, 294-296; pa- I 

j per, bow to make, 304; ware, gold '. 

I lustre for, 70. 

Stones, artificial, recipes for making dif- 
! ferent kinds of, 362-365. 

■ Strain and stress of materials, 204. 
: Strength of materials, 1'35. 

Stucco, fire-proof, for wood, 268; for 
inside walls, 301 ; Higgins', 302. 
, Superficies, mensuration of, 27, 44. 
Symbols, algebraic, 15. 

I Tallow, weight of a cubic foot of, 289. 
j Tangents, table of, 457. 
i Technical terms used in building, 331- 
! 341. 

Temperature, gradations of, 236. 237. 
i Terms used in building, 331-341. 
i Terra cotta, 268. 
Thermometrical scales, comparative ta- 
ble of the degrees of the three, 282, 
233. 
Tides, variations in, 118. 
Timber, results of experiments on the 
elasticity and strength of various 
species of, 136 ; mensuration of, 146 ; 
cause of the decay of, 805 ; methods of 
seasoning, 807-312 ; partitions, 822, 323. 
Timbers, framing of, 312; methods of 
; scarfing, 821. 

I Tin castings, bronzing liquids for, 179. 
i Toothed wheels, 68. 
Torsion, elasticity of, 143 ; resistance to, 

by shafts, 150. 
Tortoise-shell, solder for, 128; ground 

for metal, 139. 
Transverse action, resistance to, 18C; 

strength of bodies, 247. 
Trusses, or timbers framed togctlier for 
the support of weight, 320. 



Units of work, 54 

Varnishes, 89-97. 

Yegetable decomposition, circumstances 

favorable to, 305, 306. 
Vegetables, the feeding properties of 

different, 249. 
Velocity of falling bodies, 47; of wheels, 

drums, pulleys, &c., 68. 
Versed sines, table of, 415. 
Vertical pressure, resistance of bodies 

to, 141. 
Vocabulary of technical terms used in 

building, 331-341. 

Walls, preservation of. from dampness, 
153. 

Watchmaker s oil, 182. 

Water, weight of given bulks of, 63 ; ex- 
pansion of, by beat, 229 ; boiling points 
of salt, 230 ; how to find the quantity 
and weight of, in pipes of any given 
size, 280; discharge of, over weirs. 
285 

Wax,' modelling, 86. 

Wax, sealing, recipes for making dif- 
ferent kinds of, 369, 370. 

Wedge, the, 66. 

Weirs, discharge of water over, 285. 

Welding composition, 201. 

Wheel and axle, the, 65. 

Wheel, cast-iron, to find the horse-power 
it is capable of transmitting, 246. 

Wheels, velocity of, 68; toothed, ib. ; 
strength of, 152; driving, revolutions 
per mile of, 86; table of the propor- 
tions of 154; cast-iron, strength ofl 
teeth of, 127 ; dimensions of, inlictiial ! 
use, 246; proportionate strength of, in I 
horse-power, 258; toothed, table for! 
calculating the pitch of, 270; change,! 
for screw- cutting, 279. | 

Wind, speed and force of, at different 
velocities, 286. 

Wood, strength of, 23; to petrify, 70; 
fire-proof stucco for, 268; component 
parts of different varieties of, 278; 
specific gravities and other properties . 
of, 282 ; means of preventing decay in, i 
307. 

Woods, processes for staining, 218, 219; 
how to polish and preserve. 226, 227. 

Woodwork, 305, different methods of 
joining, 828-326. 

Work, units of, 54 ; of men and animals, 
55; of machines, 56; accumulated, 
56. 

Wronght-iron shafts, tiiblcs of strengths 
of, 125, 120; tensile and compressive 
strength of, 208 ; bar, transverse flexure 
of, by horizontal pressure, 218 ; experi- 
ments on transverse strength of 
rectangular tubes of, 215. 



44 



JUST PUBLISHED, 

THE 

AMERICAN COTTAGE BUILDER, 

A SERIES OF 

Designs, Plans, and Specifications, from $200 to $25,000 for Homes 
for llie People. 

BY JOHN BUJLIiOCK, 

AUTHOR OF "the RUDIMENTS OF ARCHITKCTURE AND BUILDING." 

Croivn Octavo, Price, $1,75. 

The object of this work is to furnish a practical manual of Designs for •• Homes 
for the People," from a twenty-five dollar cottage to a twenty thousand dollar 
palace, giving estimates as to cost, furnishing plans and specifications, and treat- 
ing upon the rudiments of the arts called into exercise by the practice of cottage 
building. From its practical character it must become a desirable text- book to 
the builder and mechanic, as well as to those contemplating building, either in 
ornamental or plain styles. 

This work is adapted to supply the wants of all classes in erery grade of living, 
from the humblest cottage to the stately mansion, who are interested in the erec- 
tion of beautiful and convenient houses. The estimated cost is given to each 
respective plan, so that the expenses of each building^may be accurately determ- 
ined on. Every effort has been made to add to its intrinsic value — consistently 
with its practical character— by giving it variety, in collecting all new facts relat- 
ing to recent improvements in the art of Warming and Ventilation, Garden- 
ing, &c., &c. 

The whole work will be found a " matter of fact" book on cottages and country 
residence,— a desideratum so long needed by all persons who contemplate building 
a home. 

©pinions of t^e ILeatiinfl Journals. 

This volume is a valuable contribution to the art of rural architecture. It is illustrated with a 
Lumber of excellent engravings representing various styles of cottages, and in the full details and 
clear descriptions of each style, which it gives, it will be found to be a safe guide both for taste and 
economy .---Courier and Enquirer. 

Architectural treatises of more pretension than this have frequently passed under our notice, but 
we remember few volumes of the kind that are likely to be so useful to the public. Mr. Bullock far 
nishea designs of cottages in every style, and at every cost, from $25 up to $25,000, and supplies 
minute plans and specifications, estimates as to cost, &c. He also teaches very fully and intelli 
gently the rudiments of the arts employed in cottage building.— Commercial Advertiser. 

This work contains twenty one chapters of sound sense, twenty illustrations proper, and about 
fiixty plans and illustrations of designs. Mr. Bullock is an experienced writer upon matters of arch 
itecture and art in general, and his mstructions may be firmly relied upon— drawn as they are from 
hard study and fruitful famiharity.— Buffalo Express. 

Mr. Bullock is favorably known as a clever writer on architectural subjects, and his books htiv« 
always been popular. This handsome volume is profusely illustrated with designs in every style for 
<-ottage residences. The letter-press descriptions and instructions are full, concise, and cleai,an4l 
render the treatise in our opinion useful not only to those about to erect country homes for them 
selves, but to professional builders who may be called upon to undertake such works for otbers.— 
N. Y. Express. 

This is a most useful book in this countr}'. It consists of a few plain directions for building small 
and moderate sized country houses, with a view to internal comfort, and external beauty and pro- 
•iriety. Every one who tbmks of erecting a house out of town should read tills book. It is simplr, 
end abounding with excellent hints, which may save many pounds after the house is erected. Fit 
UiT rich and poor.— Montreal Herald. 

The scope of this work is comprehensive, embracing cottages of the lowest cost, as well as build- 
mgs which might more appropriately be called palaces. It furnishes plans and specificationa for 
each, and estimates of cost. The important subject of warming and ventilating is duly considered, 
also that of drainage; and there is an instructive chapter on landscape gardening. The plates of de- 
oigns, and illustrations of the details of the designs, are very numerous, and the subjects lucidly ex 
hibited.— Journal of Commerce. 

The object of this book is to furnish those persons about to build with designs of houses costinf 
from two hundred dollars to twenty thousand, and with them valuable informal i n on the subjects or 
ventilation and gardening. Among the plates are the drawings of Prince All>errs Model Coltagas, 
and also full plans for Fowler's Octagonal House, both of them curiosities well worth examination.-- 
Boston Journal. 

One object of ♦.his truly excellent book is to encourage the spirit of impfoTetnent — to aid the cul 
tjvation r)f agricultural taste in the poor man's home, and to teach all the art of combining the useful 
•rilb tlie tasteful and elegant.--Pittsburg Token. 

Published by STRINGER & TOWNSEND, 222 Broadway, N. Y., 

And for sale by all the principal Booh sellers. 



in Indispensible Work for the Meclianie, the Engineer & the Artitt 

BOOK OF INDUSTRIAL DESIGN ; 

FORMING A COMPLETE COURSE OF 

Mechanical, Engineering & Architectural Drawing. 

TRANSLATED FROM THE FRENCH OF THE 

MESSRS. AEMENGAUD AND M. AMOUROUX, 

WITH ADDITIONS AND IMPROVEMENTS 
SIT PROFESSOR TTIIil^IAM JOHNSON. 



GENERAL CONTENTS : 

LINEAR DRAWING. — THE STUDY OF PROJECTION. — ON COLOURING SECTIONg WT-FH. AP- 
PLICATIONS. — THE INTERSECTION AND DEVELOPMENT OF SURFACES, WITH APPLICA- 
TIONS.— THE STUDY AND CONSTRUCTION OF TOOTHED GEAR.— ELEMENTARY PRINCI- 
PLES OF SHADOAVS.— APPLICATION OF SHADOWS TO TOOTHED GEAR.— THE CUTTING 
AND SHAPING OF MASONRY. — THE STUDY OF MACHINERY AND SKETCHING. — OBLIQUE 
PROJECTIONS. — PARALLEL PERSPECTIVE. — TRUE PERSPECTIVE. — EXAMPLES OF FIN 
I8HED DRAWINGS OF MACHINERY. — DRAWING INSTRUMENTS 

In«ne quarto volume^ containing 105 pages of Steel Plaies, 75 Wood Engravings, and 200 
pages of Letto-p-ess. Bound in cloth, $6.50 : in half Turkey,, $8.00. 

T?his superb work has been prepared by its distinguished Editor to meet the require- 
ments SO long felt of a comprehensive treatise on Industrial Design. His aim has been 
to present the well-known French work in a much improved English form, auid at a price 
brought within the reach of all. 

Every Principal, Mechanic, Builder, Machinist, Engineer, and Artist, has felt the want 
of a complete course of Industrial Design — not a mere collection of the first principles ol 
Geometrical Drawing, but a work that would be useful as well in Schools and Colleges 
as in the Workshop and Drawing Offices of the Engineer and Architect. 

The special mission of the Practical Draughtsman's Book of Industrial De- 
sign may almost be gathered from its title-page. It is intended to furnish gradually de- 
veloped lessons in Geometrical Drawing, applied directly to the various branches of the- 
Industrial Arts : comprehending Linear Design proper ; Isometrical Perspective, 
or the study of Projections ; the Drawing of Toothed Wheels and Eccentrics : 
with Shadowing and Colouring : Oblique Projections ; and the study of paral- 
lel and exact Perspkctive : each division being accompanied by special applications 
to the extensive ranges of Mechanics, Architecture, Foundry- Works, Carpen- 
try, Joinery, Metal Manufactures generally. Hydraulics, the construction of 
Steam Engines, and Mill-Wojik, In its compilation, the feeble attraction generally 
offered to students in elementary form ha^ been carefully considered ; and after every 
^geometrical problem, a practical example of its application has been added, to facilitate 
■Its comprehension and increase its value. A series of Plates, inarked, a, b, &c., are 
also interspersed. throughout the work, as examples of finished drawings of machinery. — 
The Letterpress relating to these Plates, together with an illustrated chapter on Drawing 
Instruments, forms an appropriate Appendix to the Volume. The general explanatory 
text embraces not only a description ol the objects and their movements, but also tables 
ai.d practical rules, more particularly those relating to the diiaensious of the principal de- 
tails of machinery, as facilitating actual construction. 

From W. Jl. Norton, Esq., Professor of Civil Enginen-ing in Yale College. 

In Uie Practical Draujbtsman we at last bave a work on Industrial design, at the same time suffi- 
ciently comprehensive in its scope and minute in its detail to meet the wants of the student of ea- 
gineerinff. My opinion of its merits is abundantly indicated by the fact that I have adopted it as a 
text-book in the School of Engineering attached to Yale College. So far as I have made use of it, I 
■fit d that it meets my exnectatioms, and doubt not that it will give general satisfaction, if introduced 
nto our Engineering ana Industrial Schools. W. A. Noktok, 

Professor of Civil Engineering in Yale College. 

From the Practical Mechanic's Journal, ( Glasgow.) 
5ircb a book in English was greatly needed. We possess for the first time m our Hteratur* n 
rtamdard work^ in many respects greatly superior to the French volume. Every page is full ofpleaa- 
uit instruction, and demonstrates competent intelligence and excessive care. The original work is 
a cteaa-book in the Industrial Schools on the continent, and the present English form of it will hold a 
high place wherever its subject is studied, either in our own lands or in the United States. It is froo 
its cheapness one of the few good books which ever3' principal should possess as a work of reference, 
Ksd which every learner should have readily accessible for constant study. 

PubUslied by STKINGER & TOWNSEND 223 Broadway, K. Y., 



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pm^^- 






D 



R 



H 



OX Pile, 



Saint John, N. B,, CMRdR. 

Of the Weight of a S:iuare Foot of Sheet Iron in pounds avoirdupois, the thickness 

being the jn umber on the Wire Gauge. 

No. 1 is 5-16ths of an inch ; No. 4, 1-4 ; No. 11, l-8th, &c. 



i 



= 14 
irk 



No. onWire Gauge 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


Pounds Avoir. ... 12*5 

1 


12 


11 


10 


9 


8 


7*5 


7 


6 


5-68 


5 


No. onWire Gauge 


12 


13 


14 


15 


16 


17 


18 


19 


20 


21 


22 


Pounds Avoir. . . . 


4-()2 


4-31 


4 


3-95 


3 


2-5 2-18 


1.93 


1-62 


1-5 


1-37 



i 



frillllllllllllllllllllllilllllllllllllllllllllilllllllllillllllllllllllillllllllilllllllllllllllllllllllllllllllllllllllllllllllllllliT: 



J. &A. McMillan, Pcinters. 



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Of the Weight of a Square Foot of Boiler Plate Iron, from 1-8 to 1 inch thick, 
in pounds avoirdupois. 



m 



5 


7-5 


4 


A 


1 


A 


i 


22-5 


1 

25 


27-5 


1 
30 


!5 

32-5 


i 

35 


37-5 


1 
In. 

40 


10 


12-5 


15 


1-75 


20 



Your Firm can Save from $4 to $5 per Ton 

BV LICENSING THIS PA TENT. 




I ^0¥IL*S P^TINT BOX PILE, 

For Nail Plate, Gas Strip, &c., &c., &c. | 

(SHOWN WITHOUT COVERS.) [ 

i > A SYSTEMATIC Method of Saving Money that Owners can understand. Is a \\ 
I 1^ PERFECT SUCCESS. ESTABLISHED by 4 YEARS' CONSTANT USE in one i [ 

OF THE Oldest and Most Experienced Firms in the Iron Manufacture S : 

OF THE United States. Write for Particulars. ^L | 

Why CAN'T You Do as Others are Doing r ^/ 



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I 



-h 



J 






^ 



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^ ^ ? 

I 



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R R ^ 



v( ^ X? 
1 ^ "^ 



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Of the Weight of Flat Bar Iron, 12 inches long, in pounds avoirdupois. 



Thickness. 


X 


3 


1. 


3. 


1 


5 


3 


1. 


1 Inch. 






8 


16 




8 


2 


8 


4 


8 






/ * 


1 
•21 ! -31 


•42 


•63 














/ ^ 
/ * 


•32 


•48 


•63 


•95 


r27 


r58 








., 




•42 


•63 


•84 


1^26 


r69 


211 


2-53 


2-96 






1 1; 


•52 1 -79 


1-05 


1^58 


2^11 


264 


3^16 


3^70 


4-22 


i 


I ^ 


•58 


•87 


1-16 


1-74 


2-32 


290- 


3-48 


4^06 


4-64 




1 l;: 


•63 


•95 


r27 


1-90 


2-53 


3^17 


3^80 


444 


5-07 


K 


1 1^' 


•74 


Ml 


1-48 


2-21 


295 


3-70 


4^43 


5^43 


5-91 


HH 


2 


•84 ^ 1-27 


1^69 


2-53 


3-38 


4-22 


5-07 


5-92 


6-76 


1— 1 


/2} 


•95 1^42 


1-90 


2-85 


3^80 


4^75 


5^70 


6-65 


7^60 


&f 


\2J 


1-06 1-58 


211 


3^17 


4^22 


5-28 


6-33 


7-40 


8^45 


g 


2| 


ri6 1-74 


232 


3-49 


4-64 


5-81 


6-97 


8-13 


9^29 






1-27 1-90 


253 


3-80 


507 


6-34 


7-60 


8-87 


10^14 




/3} 


1-37 2-06 


2-74 


4^12 


5-49 


6^86 


8^24 


10-09 


10-98 


PQ 


1 ^'' 


1-48 : 2-22 


2-95 


4^43 


5-91 


7-39 


8^87 


10-87 


11-83 




1 ^4 


1 58 : 2-38 


3-17 


4-75 


6-34 


7-92 


9-51 


11-65 


12^68 




1 4 


1-69 i 2-53 


3^38 


507 


6-76 


8-45 


10-14 


11-83 


13^52 : 




\ 4-^ 


1-90 2-85 


3-80 


5^70 


7^60 


9^50 


11-41 


13-31 


15-21 




\ 5" 


2-11 3^17 


4-22 


634 


8-45 


10-56 


12-67 


14-79 


16-90 




\6 


2-53 1 3-80 


5-07 


7-60 


10^14 


12-67 


15^21 


17-75 


20-28 



SCOVIL'S PATENT is THE ONLY COMPLETE SYSTEM OF 
Box Piling ever invented. Its value has been proved 
BY 4 Years' constant use in the "Kensington Nail and 
Rolling Mills of Messrs. Jas. Rowland & Co., No. 920 North 
Delaware Av«, Philadelphia," producing CHEAPER PLATES 
AND BETTER EDGES THAN BAR Piles. Requires NO Binders 
OR Ties. Workmen like it, as it does Not Spread and is Easily 

"^'°'"'" AO...SS E. G. SCOVIL, Mill Manager, 

CoLDBROOK P. O., Saint John, N. B. ^^i. 
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■^9IP^ 



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11 



Containing the Weight ofWrought Iron Bars, 12 inches long, in pounds avoirdupois. 



Inch. ' 


Roand. 


Stjuare. 


Inch. 


Round. 


Square. 
10-35 


Inch. 
3.V 


Round. 1 
32-52 


aiM«. 1 


} 


1-66 


•211 


If 


8-13 


41-41 




•373 


•475 


u 


9*33 


11-88 


3| 


37-34 


47-53 


* 


•664 


•845 


2 


10 62 


13'52 


4 


42-48 


5408 


1 


1-04 


1^32 


•>i 


11-99 


15-26 


4i 


47-96 


61-05 


f 


1-50 


1-90 


2} 


13-44 


17-11 


4^ 


53-77 


68-45 


t 


203 


2-59 


2* 


14-98 


19-07 


^i 


59-91 


76-27 


1 


2-65 


3-38 


21 


16-59 


21-13 


5 


66-38 


8451 


u 


3-36 


4^28 


n 


18-30 


23-29 


5i 


73-18 


9317 


H 


4-15 


5-28 


v>3 

-4 


20-08 


25-56 


5^ 


80-32 102-25 


It 


5-02 


6-39 


OJL 


21-94 


27-94 


5?- 


87-78 1 


1176 


1* 


5-99 


7-60 


3 


23-96 


30-42 


6 


95-58 ] 


21-69 


It: 


T^Ol 


8-92 


3i 


28-04 


35-70 


/ 


130-10 ] 


65'63 






HAVING had 20 years' experience in Rolling Mill and 
Nail Business, can 'be consulted on Building, Re- 
pairing, Foundations, &c., Sec. 

Estimates on Manufacturing. Prime Costs taken. 
References when required. 

E. G. SCOVIL, 

Coldbrook Post Office, St. John, N. B. 



-«^^^^l 



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A 



%"% 



<^^ ^^sm-^- 



